Geometry the area of a rectangle is x 3 + 8x 2 + 13x – 12 square units. the width of the rectangle is x + 4 units. what is the length of the rectangle?

To find the length of the rectangle, we need to divide the area of the rectangle by its width. The area is given by the polynomial \( x^3 + 8x^2 + 13x - 12 \) square units, and the width is given as \( x + 4 \) units.

The length (L) can be calculated as follows:

\( L = \frac{\text{Area}}{\text{Width}} \)

Plugging in the values we have:

\( L = \frac{x^3 + 8x^2 + 13x - 12}{x + 4} \)

We can find the length by performing polynomial division:

x^2 + 4x + 1 _____________________ x + 4 ) x^3 + 8x^2 + 13x - 12 - (x^3 + 4x^2) _______________ 4x^2 + 13x - (4x^2 + 16x) ______________ -3x - 12 + (3x + 12) ______ 0

Each step of the polynomial long division process subtracts multiples of \( x + 4 \) from the parts of the area's polynomial, breaking it down step by step until the remainder is zero.

In summary, the length of the rectangle is:

\( L = x^2 + 4x + 1 \)