Question

1. 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?

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). So, to find the length \( l \), we can use the formula \( l=\frac{A}{\text{width}} \). Here, the area \( A = x^{3}+8x^{2}+13x - 12 \) and the width \( w=x + 4 \). So we need to perform polynomial long division or factor the cubic polynomial and divide by \( x + 4 \).

Step2: Factor the cubic polynomial \( x^{3}+8x^{2}+13x - 12 \)

We can use synthetic division with root \( x=-4 \) (since the width is \( x + 4 \), so we test \( x=-4 \) as a root).

Set up synthetic division:
\[

$$\begin{array}{r|rrrr} -4 & 1 & 8 & 13 & -12 \\ & & -4 & -16 & 12 \\ \hline & 1 & 4 & -3 & 0 \\ \end{array}$$

\]
The coefficients of the quotient polynomial are \( 1,4,-3 \), so the cubic polynomial factors as \( (x + 4)(x^{2}+4x-3) \)? Wait, no, wait. Wait, the remainder is 0, so \( x^{3}+8x^{2}+13x - 12=(x + 4)(x^{2}+4x-3) \)? Wait, no, let's check the multiplication: \( (x + 4)(x^{2}+4x - 3)=x^{3}+4x^{2}-3x+4x^{2}+16x - 12=x^{3}+8x^{2}+13x - 12 \). Yes, that's correct.

Step3: Find the length

Since \( A=\text{length}\times\text{width} \), and \( A=(x + 4)(x^{2}+4x - 3) \), and \( \text{width}=x + 4 \), then the length \( l=\frac{(x + 4)(x^{2}+4x - 3)}{x + 4} \). Assuming \( x
eq - 4 \) (since width can't be zero), we can cancel out \( x + 4 \) from numerator and denominator. So the length is \( x^{2}+4x - 3 \).

Answer:

The length of the rectangle is \( \boldsymbol{x^{2}+4x - 3} \) units.