a rectangle with width 2x+5 inches has an area of 2x^4+11x^3-5x^2-98x-120 square inches. Write a polynomial that represents its lenght

Why did the rectangle go on a diet? Because it wanted to reduce its "width" and become "lengthy"!

Now, let's find the length of the rectangle. The area of a rectangle is given by multiplying its length and width. So, we can use the given information to solve for the length.

Area = Length * Width

2x^4 + 11x^3 - 5x^2 - 98x - 120 = Length * (2x + 5)

To find the polynomial that represents the length, we need to isolate it on one side of the equation. Let's solve for Length:

Length = (2x^4 + 11x^3 - 5x^2 - 98x - 120) / (2x + 5)

Therefore, the polynomial that represents the length of the rectangle is:

Length = (2x^4 + 11x^3 - 5x^2 - 98x - 120) / (2x + 5)

To find the length of the rectangle, we can use the formula for the area of a rectangle, which is given by:

Area = Length × Width

In this case, the area of the rectangle is represented by the polynomial 2x^4+11x^3-5x^2-98x-120 square inches, and the width is given as 2x+5 inches.

We can set up an equation using the formula:

2x^4+11x^3-5x^2-98x-120 = Length × (2x+5)

To find the polynomial representation of the length, we need to solve for Length.

Let's solve the equation step by step:

1. Distribute the Length on the right side of the equation:
2x^4+11x^3-5x^2-98x-120 = 2xLength + 5Length

2. Rearrange the equation:
2x^4+11x^3-5x^2-98x-120 - 2xLength - 5Length = 0

3. Group like terms:
2x^4 + (11x^3 - 2xLength) + (-5x^2 - 98x - 5Length) - 120 = 0

Now, we have grouped the common terms to create a polynomial representation of the length:

Length = -5x^2 - 98x - 5Length + 120 - 11x^3 + 2xLength - 2x^4

Simplifying the polynomial further if possible depends on the given information or any additional instructions.

To find the length of the rectangle, we can use the formula for the area of a rectangle:

Area = length * width

Given that the area is 2x^4 + 11x^3 - 5x^2 - 98x - 120 square inches, and the width is 2x + 5 inches, we can set up the equation as follows:

2x^4 + 11x^3 - 5x^2 - 98x - 120 = length * (2x + 5)

Now we will solve for the length by dividing both sides of the equation by (2x + 5):

(2x^4 + 11x^3 - 5x^2 - 98x - 120) / (2x + 5) = length

To simplify the expression, we can use polynomial long division or synthetic division. However, it seems the given polynomial is quite complex, so we'll use synthetic division for simplicity.

Applying synthetic division, we have:

-5 | 2 11 -5 -98 -120
------------------------------------
2 -9 40 -98 412

The result of the synthetic division is 2x^3 - 9x^2 + 40x - 98 with a remainder of 412.

Therefore, the polynomial that represents the length of the rectangle is 2x^3 - 9x^2 + 40x - 98.