Ponderosa Paint and Glass carries three brands of paint. A customer wants to buy another gallon of paint to match paint she purchased at the store previously. She can't recall the brand name and does not wish to return home to find the old can of paint. So she selects two of the three brands of paint at random and buys them. Her husband also goes to the paint store and fails to remember what brand to buy. So he also purchases two of the three brands of paint at random. Determine the probability that both the woman and her husband fail to get the correct brand of paint. (Hint: Are the husband's selections independent of his wife's selections

Answer:

The slope is = -1/4

Step-by-step explanation:

We are looking for the slope of the line CD. You are given the coordinates for each point as follows:

A(0,-1)     B(5,3)    C(2,3)   D(6,2)

To find the slope of a straight line, you use the formula:

slope (m) = (y2 - y1)/(x2 - x1)

In other words the slope is the "rise over run":

Remember: the coordinates of a point are (x;y). Choose one of the points to be point 1 and the other to be point 2. I would suggest that you follow the order they have given you ie. CD:

So if our points are C (2; 3) and D (6, 2) we can say point C is point 1 and point D is point 2. Plug into the formula:  

slope (m) = (y2-y1)/(x2-x1)

y1  = 3

y2 = 2

x1  = 2

x2 = 6

∴ m = (2-3)/(6-2)

      = -1/4

Answer:

(f+g)(3) = 347 calories

Step-by-step explanation:

f(x) = 2x + 210

g(x) = 2x + 125

(f+g)(x) is the sum of the functions

(f+g)(x) = 2x+210 + 2x+125

Combining like terms

           =4x+335

Now let x = 3

(f+g)(3) = 4*3 +335

            = 12 +335

            = 347

(f+g)(3) = 347 calories

Answer:

347

Step-by-step explanation:

f(x) = 2x + 210

g(x) = 2x + 125

we know that (f+g)(x)=f(x)+g(x)

so (f+g)(x)= 2x + 210 + 2x + 125 = 4x + 335

(f+g)(3) simply means that x=3

therefore

(f+g)(3) = 12 + 335 = 347

Answer:

55.96 miles

Step-by-step explanation:

Distance = Speed X time

Car first moved east for 1 hr , distance = 40 x 1 = 40 miles

Car then moved north-east for 30 min, distance = 40 x 0.5 = 20 miles

From trigonometry,

sin 45 = [tex]\frac{y}{20}[/tex] ; y = 20 sin 45 ; y= 14.14 miles

cos 45 = [tex]\frac{x}{20}[/tex] ; x = 20 cos 45 ; x = 14.14 miles

using Pythagoras theorem;

[tex]z^{2}[/tex] = [tex]y^{2}[/tex] + [tex](40+x)^{2}[/tex]

    = [tex]14.14^{2}[/tex] + [tex](40+14.14)^{2}[/tex]

     = 3131.08

[tex]z^{2}[/tex] = [tex]\sqrt{3131.08}[/tex]

z = 55.96 miles

Answer:

0.40

Step-by-step explanation:

to find out the probability that at least one of a pair of fair dice lands of 5, given that the sum of the dice is 8

Let A = sum of dice is 8

B = one lands in 5

P(B/A) = P(AB)/P(A) by conditional probability

P(AB) = sum is 8 and one is 5

So (5,3) or (3,5)

P(A) = sum  is 8.

i.e. (2,6) (2,6) (3,5) (5,3) (4,4)

Required probability

= n(AB)/n(A)

=[tex]\frac{2}{5} =0.40[/tex]

Answer:

t = 460.52 min

Step-by-step explanation:

Here is the complete question

Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 200 liters of a dye solution with a concentration of 1 g/liter. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 2 liters/min, the well-stirred solution flowing out at the same rate.Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value.

Solution

Let Q(t) represent the amount of dye at any time t. Q' represent the net rate of change of amount of dye in the tank. Q' = inflow - outflow.

inflow = 0 (since the incoming water contains no dye)

outflow = concentration × rate of water inflow

Concentration = Quantity/volume = Q/200

outflow = concentration × rate of water inflow = Q/200 g/liter × 2 liters/min = Q/100 g/min.

So, Q' = inflow - outflow = 0 - Q/100

Q' = -Q/100 This is our differential equation. We solve it as follows

Q'/Q = -1/100

∫Q'/Q = ∫-1/100

㏑Q  = -t/100 + c

[tex]Q(t) = e^{(-t/100 + c)} = e^{(-t/100)}e^{c} = Ae^{(-t/100)}\\Q(t) = Ae^{(-t/100)}[/tex]

when t = 0, Q = 200 L × 1 g/L = 200 g

[tex]Q(0) = 200 = Ae^{(-0/100)} = Ae^{(0)} = A\\A = 200.\\So, Q(t) = 200e^{(-t/100)}[/tex]

We are to find t when Q = 1% of its original value. 1% of 200 g = 0.01 × 200 = 2

[tex]2 = 200e^{(-t/100)}\\\frac{2}{200} = e^{(-t/100)}[/tex]

㏑0.01 = -t/100

t = -100㏑0.01

t = 460.52 min

Multiply the price of popcorn by the number of bags so 6b.

Multiply the price of candy by the number bought, so 3.25c

Add those together to get total :

6b + 3.25c

Now the most she can spend is 50 so set the equation to less than it equal to what she can spend:

6b + 3.25c <= 50

Final answer:

The inequality representing Samantha's budget for purchasing popcorn and candies is 6b + 3.25c ≤ 50.

Explanation:

To formulate the inequality representing the possible values for the number of bags of popcorn (b) and the number of candies (c) Samantha can purchase without exceeding her $50 budget, we can set up an expression based on the given prices for each item.

The cost of the popcorn bags is $6, and the cost of each candy is $3.25.

The inequality that represents this scenario would be:

6b + 3.25c ≤ 50

This inequality denotes that the total cost of b popcorn bags and c candies, which is the sum of $6 times the number of popcorn bags plus $3.25 times the number of candies, should be less than or equal to $50.

Samantha must choose combinations of b and c that satisfy this inequality to stay within her budget.

Answer: the shortest side is 10 centimeters.

The length of each of the other sides is 10 centimeters each.

Step-by-step explanation:

Let x represent the length of the shortest side of the triangle.

Two sides of the triangle are each twice as long as the shortest side. This means that the length of the two sides would be 2x.

The perimeter of a triangle is the sum of each side of the triangle.

The puzzle piece in the shape of a triangle has perimeter 25 centimeters. This means that

x + 2x + 2x = 25

5x = 25

x = 25/5

x = 5

The length of each of the two sides is

2x = 2 × 5 = 10

The length of the shortest side of the triangle is 5 centimeters.

Let's denote the length of the shortest side of the triangle as "x" centimeters. According to the problem, the other two sides are each twice as long as the shortest side. Therefore, the lengths of the other two sides are "2x" centimeters each.

Now, we can use the information given about the perimeter to set up an equation:

Perimeter = Sum of all sides

Given that the perimeter is 25 centimeters:

25 = x + 2x + 2x

Now, combine like terms on the right side:

25 = 5x

To solve for x, divide both sides by 5:

x = 25 / 5

x = 5

So, the length of the shortest side of the triangle is 5 centimeters.

Learn more about length here:

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Answer:

Step-by-step explanation:

Answer:

576 cm²

Step-by-step explanation:

Given:

Number of sections in a suncatcher (n) = 6

Each section is in the shape of a parallelogram.

Base of parallelogram = 12 cm

Height of parallelogram = 8 cm

Now, area of a parallelogram is given as:

Area of parallelogram = Base × Height

Area of 1 parallelogram = 12 cm × 8 cm = 96 cm²

Now, there are 6 parallelogram shaped sections.

So, area of the sections = area of 1 section × total number of sections

∴ Area of 6 sections = 96 cm² × 6 = 576 cm²

Therefore, the total area of the sections of a suncatcher is 576 cm².

Yo sup??

This question can be solved by applying trigonometric ratios

let angle U be x, then

tanx=4/7

x=29.7

Hope this helps

Answer:

Therefore [tex](\frac{46}{59},-\frac{62}{59} )[/tex] is a solution of given liner equations.

Step-by-step explanation:

Given system of equation are

7x+9y=-4..............(1)

5x-2y=6...............(2)

Equation (1)×5 - equation (2)×7

35x +45y-(35x-14y)= -20-42

⇔ 35x+45y-35x+14y = -62

⇔59 y = -62

[tex]\Leftrightarrow y =-\frac{62}{59}[/tex]

Putting the value of y in equation (1)

[tex]7x +9.(-\frac{62}{59} )= -4[/tex]

[tex]\Leftrightarrow 7x += -4+\frac{558}{59}[/tex]

[tex]\Leftrightarrow 7x = \frac{322}{59}[/tex]

[tex]\Leftrightarrow x =\frac{322}{59\times 7}[/tex]

[tex]\Leftrightarrow x =\frac{46}{59}[/tex]

Therefore  [tex]x =\frac{46}{59}[/tex] and  [tex]y =-\frac{62}{59}[/tex]

Therefore [tex](\frac{46}{59},-\frac{62}{59} )[/tex] is a solution of given liner equations.

Answer:

D.)He made a mistake in his calculations when substituting the ordered pair into the equation 5x – 2y = 6 and simplifying.

Step-by-step explanation:

I just got this right for the exam review on edge

(1) Vertical asymptote: [tex]x=-4[/tex]

(2) Domain: [tex]x>-4[/tex]

(3) X intercept: [tex](-2,0)[/tex] and Y intercept : [tex](0,1)[/tex]

(4) The function g(x) is shifted 4 units to the left and shifted 1 unit down.

Explanation:

The parent function is [tex]f(x)=\log _{2} x[/tex]

The transformed function is [tex]g(x)=\log _{2}(x+4)-1[/tex]

(1) Vertical asymptote:

The vertical asymptote of a function can be determined by equating

[tex]x+4=0[/tex]

Thus, [tex]x=-4[/tex]

The vertical asymptote is [tex]x=-4[/tex]

(2) Domain:

The domain of a function is the set of all independent x-values.

[tex]x+4>0[/tex]

Thus, [tex]x>-4[/tex]

The domain of a function is [tex]x>-4[/tex]

(3) X and Y intercepts:

To determine the x intercept, let us substitute y=0 in [tex]g(x)=\log _{2}(x+4)-1[/tex]

[tex]\begin{equation}\begin{aligned}\log _{2}(x+4)-1 &=0 \\\log _{2}(x+4) &=1 \\x+4 &=2^{1} \\x &=-2\end{aligned}[/tex]

Thus, the x intercept is [tex](-2,0)[/tex]

To determine the y intercept, let us substitute x=0 in [tex]g(x)=\log _{2}(x+4)-1[/tex]

[tex]\begin{equation}\begin{aligned}y &=\log _{2}(0+4)-1 \\&=\log _{2} 4-1 \\&=2-1 \\&=1\end{aligned}[/tex]

Thus, the y intercept is [tex](0,1)[/tex]

(4) To determine the transformation:

The transformed function [tex]g(x)=\log _{2}(x+4)-1[/tex] is shifted 4 units to the left and shifted 1 unit downwards.

Step-by-step explanation:

The given geometric sequence:

- 8, - 40, - 200, - 1000

Here, first term(a) = - 8, common ration(r) = [tex]\dfrac{-40}{-8}[/tex] = 5

To find, an explicit formula for the given geometric sequence = ?

We know that,

The explicit formula for the geometric sequence

[tex]a_{n} =ar^{n-1}[/tex]

∴ An explicit formula for the given geometric sequence

[tex]a_{n} =(-8)(5)^{n-1}[/tex]

[tex]a_{n} =\dfrac{-8}{5} (5)^{n}[/tex]

∴ An explicit formula for the given geometric sequence, [tex]a_{n} =\dfrac{-8}{5} (5)^{n}[/tex]

Answer:

what the guy above me said

Step-by-step explanation:

Answer:

a negative correlation between the temperature and the amount of snow still on the ground

a negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left

Explanation:

When it comes to "correlations," a negative one refers to an "inverse" relationship between two variables. So, this means that as one variable increases, the other decreases and vice-versa.

The question is asking for two options that are "casual" (common) when it comes to "negative correlation." So, the answers are:

a negative correlation between the temperature and the amount of snow still on the ground

a negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left

Answer:

Step-by-step explanation:

These are the two relationships that are most likely to be causal. Causal relationships are those in which one aspect is the cause of the second one being described. Moreover, a negative correlation is one in which one aspect of the relationship increases while the other one decreases. In the first example, as the temperature goes up, this causes the amount of snow on the ground to go down. In the second example, as you upload more pictures to a website, there is less storage space.

Answer:

39 °F

Step-by-step explanation:

There's 4 hours difference between given times so the change in the temperature would be 4 × 8 = 32° F if the temperature drops 8 °F every hour therefore the temperature would be 71 - 32 = 39 °F

x = 37.5 (or) [tex]\frac{75}{2}[/tex]

Solution:

Given [tex]\triangle A B C \sim \triangle D B E[/tex].

Let us take BE = x and BC = 25 + x.

To determine the value of x:

If two triangles are similar then the corresponding angles are congruent and the corresponding sides are in proportion.

[tex]$\frac{AC}{DE}=\frac{B C}{B E}[/tex]

[tex]$\frac{50}{30} =\frac{25+x}{x}[/tex]

Do cross multiplication, we get

[tex]50x=30(25+x)[/tex]

[tex]50x=750+30x[/tex]

Subtract 30x from both sides of the equation.

[tex]20 x=750[/tex]

Divide by 20 on both sides of the equation, we get

x = 37.5 (or) [tex]\frac{75}{2}[/tex]

Hence the value of x is 37.5 or [tex]\frac{75}{2}[/tex].

Answer: the number of adult and student tickets sold are (45, 29)

Step-by-step explanation:

Let x represent the number of adult tickets that were sold.

Let y represent the number of student tickets that were sold.

On Wednesday, a total of 74 tickets were sold. This means that

x + y = 74

x = 74 - y- - - - - - - - - - - - - - 1

If adult tickets are sold for $15 and student tickets are sold for $11 and the money collected was $994, it means that

15x + 11y = 994- - - - - - - - - - - - - - 2

Substituting equation 1 into equation 2, it becomes

15(74 - y) + 11y = 994

1110 - 15y + 11y = 994

- 15y + 11y = 994 - 1110

- 4y = - 116

y = - 116/ - 4

y = 29

Substituting y = 29 into equation 1, it becomes

x = 74 - 29 = 45

The number of adult tickets and student tickets sold is (x, y) = (45, 29)

To determine the number of adult tickets x and student tickets y sold, we need to solve the system of equations given by the following conditions:

1. The total number of tickets sold is 74:

x + y = 74

2. The total revenue from the tickets is $994, with adult tickets sold for $15 each and student tickets sold for $11 each:

15x + 11y = 994

We can solve this system of equations using the substitution or elimination method. Here, we will use the elimination method.

First, let's write the two equations clearly:

x + y = 74

15x + 11y = 994

To eliminate one of the variables, we can multiply the first equation by 11, so the coefficients of  y in both equations will be the same:

[tex]\[ 11(x + y) = 11 \cdot 74 \][/tex]

11x + 11y = 814

Now we have:

11x + 11y = 814

15x + 11y = 994

Next, we subtract the first modified equation from the second equation to eliminate y

(15x + 11y) - (11x + 11y) = 994 - 814

15x + 11y - 11x - 11y = 180

4x = 180

x = 45

Now, substitute x = 45 back into the first original equation to find y

x + y = 74

45 + y = 74

y = 29

Therefore, the number of adult tickets and student tickets sold is:

(x, y) = (45, 29)

Answer:

47 ducks and 7 cows

Step-by-step explanation:

Ducks have 2 legs and cows have 4 legs.

If all the animals are ducks,

number of legs= 54(2)= 108

Difference in number of legs= 122 -108= 14

Difference in no. of legs per animal= 4 -2 = 2

Number of cows=14 ÷2 = 7

Number of ducks= 54-7= 47

Thus, he have 47 ducks and 7 cows.

Let's check:

Total number of feet= 7(4)+ 47(2) = 122 ✓

You could also first assume that all the animals are cows. The answer would be the same.

Answer: D

Step-by-step explanation:

it makes the most sense ( do ont know how to explain but I do know it is right. )

Answer:

D.) No, product means multiplication, and the student wrote a division expression.

Step-by-step explanation:

did the assignment. good luck! <3

Answer:

a=128 feet   b=256 feet

Step-by-step explanation:

hope i got it

Answer:

A.) 128FT

B.) A=256

Step-by-step explanation:

A.)

16 x 4 = 64FT

62 x 2 = 128FT

So, 2 Laps Would Equal Up To 128FT.

B.)

Find the length of one of the sides (as each side is the same length) and then multiply this by itself to find the area.

Answer:

Option A:        [tex]$ \sum_{n = 1}^{15} {\textbf{- 9 + 6 n}} $[/tex]

Step-by-step explanation:

We are given with the series - 9  - 3  + 3   +  9 +  . . . .  +  81.

Note that the second term is obtained by adding 6 to the first term.

Each consecutive term is obtained by adding 6 to its previous term.

Therefore, we should be adding six two times to get the third term from the first term.

Putting it Mathematically, we get: - 9 + 6n

This gives all the terms of the sequence. Since, we have to add all the terms we take the summation.

Also, note that 81 is the 15 th term.

Therefore, - 9  - 3  + 3   +  9 +  . . . .  +  81 =    [tex]$ \sum_{n = 1}^{15} {- 9 + 6 n} $[/tex]

Hence, the answer.

Final answer:

The sequence -9, -3, 3, 9, ..., 81 increases by 6 each time and follows the arithmetic sequence formula -9 + 6(n-1), which simplifies to -15 + 6n. The summation notation that represents this sequence is the sum of -15 + 6n from n = 0 to 15.

Explanation:

The given sequence is -9, -3, 3, 9, ..., 81, and we can see that it is increasing by 6 each time (an arithmetic sequence). The nth term of an arithmetic sequence can be written as a + (n-1)d, where a is the first term and d is the common difference. Here, the first term a is -9 and the common difference d is 6. Plugging these values into the formula gives us the nth term as -9 + 6(n-1). We can simplify this to -9 + 6n - 6, which further simplifies to -15 + 6n.

To find the last term which is 81, we can set the nth term equal to 81: -15 + 6n = 81. Solving for n gives us n = 16, meaning there are 15 terms before it (since we start counting from n = 0).

The summation notation for the given sequence is then the sum of the terms from n = 0 to n = 15 of the general term -15 + 6n.

The required function that represents the situation is [tex]y=800(\frac{1}{4} )^t[/tex]

The standard exponential function is expressed as:

[tex]y=ab^x[/tex] where:

Given the following parameters:

a = 800

b = 1/4

Substitute into the formula to have;

[tex]y=800(\frac{1}{4} )^t[/tex]

Hence the required function that represents the situation is [tex]y=800(\frac{1}{4} )^t[/tex]

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Answer:

It's 800(3/4)^t

Step-by-step explanation:

Answer:

In a study to determine whether how long a student sleeps affects test scores, the independent variable is the length of time spent sleeping while the dependent variable is the test score. You want to compare brands of paper towels, to see which holds the most liquid.

hopes it helps

Answer:

The approximate length of the actual car is 43 feet.

Step-by-step explanation:

Given:

The scale on a model railroad train set is 3.5 millimeters to 1 foot. If the length of a model car in the set is 150 millimeters.

Now, to find the approximate length of the actual car.

Let the actual length of the car is [tex]x.[/tex]

The length of a model car in the set is 150 millimeters.

As, given the scale on a model railroad train set is 3.5 millimeters to 1 foot.

So, 3.5 millimeters equivalent to 1 foot.

Thus, 150 millimeters to [tex]x[/tex].

Now, to get the actual length of the car by using cross multiplication method:

[tex]\frac{3.5}{1} =\frac{150}{x}[/tex]

By using cross multiplying we get:

[tex]3.5x=150[/tex]

Dividing both sides by 3.5 we get:

[tex]x=42.85\ feet.[/tex]

The approximate length of the actual car = 43 feet.

Therefore, the approximate length of the actual car is 43 feet.

Answer:

43ft

Step-by-step explanation:

The answer is B.

Here's another way to solve this problem.

1. Write the scale ratio

3.5 mm

1 ft

2.  Choose a variable, such as r, to represent the approximate length of the actual railroad car

3. Write the ratio of the length of the model car to the length of the actual car:

150 mm

r ft

4. Write a proportion

3.5=150

1= r

5. Cross multiply

3.5

r=150

   3.5

r=42.9, which rounds to 43.

so the, approximate length of the actual car is 43 feet

Answer:

ARITHMETIC SENTENCE

Step-by-step explanation:

Final answer:

Naledi's altitude sequence is an arithmetic sequence because it increases by a constant amount each hour.

Explanation:

Naledi's altitude at the beginning of the nth hour of her climb can be described by the function g(n), where g(n) is the altitude at that hour. Since she starts at 40 meters above sea level and increases her altitude by 10 meters every hour, the sequence representing her altitude over time is an arithmetic sequence. This is because each term in the sequence increases by a constant value, which is the definition of an arithmetic sequence.

The general formula for the nth term of an arithmetic sequence is aₙ = a₁ + (n - 1)d, where a₁ is the first term and d is the common difference between terms. In Naledi's case, a₁ is 40 meters (her initial altitude), and d is 10 meters (the constant increase each hour).

Answer:

[tex]p(x) = {x}^{4} + {x}^{3} - 7 {x}^{2} - x + 6[/tex]

Step-by-step explanation:

The polynomial function has zeros

x=1, x=2,x=-3,x=-1

This means the factored form of the polynomial is

[tex]p(x) = (x - 1)(x + 1)(x + 3)(x - 2)[/tex]

We expand to get:

[tex]p(x) = ( {x}^{2} - 1)( {x}^{2} + x - 6)[/tex]

We expand further to get:

[tex]p(x) = {x}^{2}( {x}^{2} + x - 6) - 1({x}^{2} + x - 6)[/tex]

[tex]p(x) = {x}^{4} + {x}^{3} - 6 {x}^{2} - {x}^{2} - x + 6[/tex]

This simplifies to:

[tex]p(x) = {x}^{4} + {x}^{3} - 7 {x}^{2} - x + 6[/tex]

This is the standard form of the polynomial since it is written in descending powers of x.

Answer:

3hrs 12 mins

Step-by-step explanation:

For every 10% charge she gets 1hr 20mins (or 80 mins)

Charge drop = (81% - 57%) = 24%

Since 10% gives 80 mins;

          24% gives (24/10) x 80 mins = 192 mins

since 1hr = 60 mins

192 mins = 192/60 = 3hrs 12 mins

Final answer:

Mandi played the game on her phone for 3 hours and 12 minutes, as she experienced a 24% drop in battery charge, with each 10% equating to 1 hour and 20 minutes of playtime.

Explanation:

The student's question involves calculating the duration of time Mandi played a game on her phone based on the percentage drop in battery charge. Mandi’s phone loses 10% charge for every 1 hour and 20 minutes of use. The charge dropped from 81% to 57%, which is a 24% drop. To find out how long she played the game, we calculate the number of 10% intervals in 24% and then multiply by the duration of one interval.

First, we determine how many 10% intervals are in 24%, which is calculated as 24% ÷ 10% = 2.4 intervals. Each interval equates to 1 hour and 20 minutes (which is the same as 80 minutes). Therefore, Mandi played for 2.4 intervals × 80 minutes per interval:

2.4 intervals × 80 minutes/interval = 192 minutes.

To convert minutes into hours and minutes, we divide by 60:

192 minutes ÷ 60 minutes/hour = 3 hours and 12 minutes.

So, Mandi played the game for 3 hours and 12 minutes.

Answer: $4,884.75

Step-by-step explanation:

The formula for determining simple interest is expressed as

I = PRT/100

Where

I represents interest paid on the amount deposited.

P represents the principal or amount deposited.

R represents interest rate

T represents the duration of the savings in years.

From the given information,

P = 4500

R = 2.85

T = 3 years

I = (4500 × 2.85 × 3)/100 = $384.75

The total amount that they will have in savings after 3 years is

4500 + 384.75 = $4884.75

(3x³ - 17x² + 15x - 25)/(x - 5) =

= (3x³ - 15x² - 2x² + 0x + 5x - 25)/(x - 5)

= [3x²(x - 5) - 2x(x - 5) + 5(x - 5)]/(x - 5)

= (3x² - 2x + 5)(x - 5)/(x - 5)

= 3x² - 2x + 5

(5x³ + 18x² + 7x - 6)/(x + 3) =

= (5x³ + 5x² + 13x² + 13x - 6x - 6³)/(x + 3)

= [5x²(x + 1) + 13x(x + 1) - 6(x + 1)]/(x + 3)

= (x + 1)(5x² + 13x - 6)/(x + 3)

= (x + 1)(5x² + 15x - 2x - 6)/(x + 3)

= (x + 1)[5x(x + 3) - 2(x + 3)]/(x + 3)

= (x + 1)(5x - 2)(x + 3)/(x + 3)

= (x + 1)(5x - 2)

(4x³ + 8x² - 9x - 18)/(x + 2) =

= [4x²(x + 2) - 9(x + 2)]/(x + 2)

= (4x² - 9)(x + 2)/(x + 2)

= (4x² - 9)

= (2x - 3)(2x + 3)

(9x³ - 16x - 18x² + 32)/(x - 2) =

= [x(9x² - 16) - 2(9x² - 16)]/(x - 2)

= (9x² - 16)(x - 2)/(x - 2)

= 9x² - 16

= (3x - 4)(3x + 4)

(- x³ + 75x - 250)/(x + 10) =

= ( - x³ + 5x² - 5x² + 25x + 50x - 250)/(x + 10)

= [ - x²(x -5) - 5x(x - 5) + 50(x - 5)]/(x + 10)

= - (x - 5)(x² + 10x - 5x - 50)/(x + 10)

= - (x - 5)[x(x + 10) - 5(x + 10)]/(x + 10)

= - (x - 5)(x - 5)(x + 10)/(x + 10)

= - (x - 5)²

(3x³ - 16x² - 72)/(x - 6) =

= (3x³ - 18x² + 2x² - 72)/(x - 6)

= [3x²(x - 6) + 2(x² - 36)]/(x - 6)

= [3x²(x - 6) + 2(x - 6)(x + 6)]/(x - 6)

= [3x² + 2(x + 6)](x - 6)/(x - 6)

= 3x² + 2x + 12