please help me with this question and check my answers.

Solve the problem.

A rectangle with width 2x + 5 inches has an area of 2x4 + 9x3 - 12x2 - 79x - 60 square inches. Write a polynomial that represents its length. ( I don't know how to solve this problem)

Use synthetic division and the Remainder Theorem to find the indicated function value.

f(x) = x^4 + 4x^3 + 8x^2 - 5x + 3; f(-2)
I GOT 29

f(x) = 4x^3 - 3x^2 - 3x + 20; f(-2)
i got -18

f(x) = 6x^4 + 9x^3 + 5x^2 - 6x + 24; f(-3)
i got 330

Use synthetic division to divide f(x) = x^3 + 7x^2 - 6x - 72 by x + 4. Use the result to find all zeros of f.
-i got {-4,-6,3}

Use synthetic division to show that the number given to the right of the equation is a solution of the equation, then solve the polynomial equation.

x^3 + 5x^2 + 2x - 8 = 0; -2
i got {1,-4,-2}

For the first one, since Area = LxW

and W = 2x+5 then
L = (2x^4 + 9x^3 - 12x^2 - 79x - 60)/(2x+5)

the rest of your answers are correct

(x^3+mx^2+nx-3)(x-2)(x+1)

To solve the problem of finding the polynomial representing the length of a rectangle, we need to use the formula for the area of a rectangle, which is length times width.

Given that the width is 2x + 5 inches, and the area is 2x^4 + 9x^3 - 12x^2 - 79x - 60 square inches, we can set up the equation:

Length * Width = Area

L * (2x + 5) = 2x^4 + 9x^3 - 12x^2 - 79x - 60

Now, we need to solve for L, the length, by isolating it on one side of the equation:

2xL + 5L = 2x^4 + 9x^3 - 12x^2 - 79x - 60

Combine like terms on the right side:

2xL + 5L = 2x^4 + 9x^3 - 12x^2 - 79x - 60

To solve for L, we need to factor out the common factor of x in the right side:

L(2x + 5) = x * (2x^3 + 9x^2 - 12x - 79) - 60

Divide both sides by (2x + 5) to isolate L:

L = (x * (2x^3 + 9x^2 - 12x - 79) - 60) / (2x + 5)

This is the polynomial that represents the length of the rectangle.

Now let's move on to the questions about using synthetic division and the Remainder Theorem.

For the first question, we have the polynomial f(x) = x^4 + 4x^3 + 8x^2 - 5x + 3 and we need to find f(-2).

To do this, we substitute -2 into the polynomial:

f(-2) = (-2)^4 + 4(-2)^3 + 8(-2)^2 - 5(-2) + 3

Calculating this expression, we get:

f(-2) = 16 - 32 + 32 + 10 + 3

Simplifying, we find:

f(-2) = 29

So the correct answer is indeed 29.

Similarly, for the second question, we have the polynomial f(x) = 4x^3 - 3x^2 - 3x + 20 and we need to find f(-2).

Substituting -2 into the polynomial, we get:

f(-2) = 4(-2)^3 - 3(-2)^2 - 3(-2) + 20

Evaluating this expression, we find:

f(-2) = -32 - 12 + 6 + 20

Simplifying, we obtain:

f(-2) = -18

Hence, your answer of -18 is correct.

Moving on to the third question, we have the polynomial f(x) = 6x^4 + 9x^3 + 5x^2 - 6x + 24 and we need to find f(-3).

Substituting -3 into the polynomial, we get:

f(-3) = 6(-3)^4 + 9(-3)^3 + 5(-3)^2 - 6(-3) + 24

Evaluating this expression, we find:

f(-3) = 6(81) - 9(27) + 5(9) + 18 + 24

Simplifying further, we obtain:

f(-3) = 486 - 243 + 45 + 18 + 24

Calculating this expression, we find:

f(-3) = 330

So your answer of 330 is correct.

Now let's move on to the next question, which is about synthetic division and finding zeros of a polynomial.

We are given the polynomial f(x) = x^3 + 7x^2 - 6x - 72 and we need to divide it by x + 4 using synthetic division.

We set up the synthetic division as follows:

-4 | 1 7 -6 -72
___________________


We bring down the coefficient of the highest degree term, which is 1:

-4 | 1 7 -6 -72
__________________

1

Next, we multiply -4 by 1 and write the result under the next coefficient:

-4 | 1 7 -6 -72
___________________
-4

We add the numbers in the second column:

-4 | 1 7 -6 -72
___________________
-4
________
1

Next, we multiply -4 by the sum we just obtained, which is 1, and write the result under the next coefficient:

-4 | 1 7 -6 -72
___________________
-4
________
1
12

We continue this process until we have gone through all the coefficients:

-4 | 1 7 -6 -72
___________________
-4
________
1
12
________
1 8
-32
_________
1 8 -38
6
___________
1 8 -38 -6

The result of the synthetic division is 1x^2 + 8x - 38 with a remainder of -6.

To find the zeros of f(x), we set the expression inside the synthetic division equal to zero:

1x^2 + 8x - 38 = 0

Now we need to solve this quadratic equation to find the zeros. You can use the quadratic formula or factoring to solve it.