Question

a rectangle has an area of x^3 + 3x^2 - 18x square inches and a width of x + 6 inches. what is the length in inches? show all work by hand

Explanation:

Step1: Recall area formula

The area formula for a rectangle is $A = lw$, where $A$ is the area, $l$ is the length and $w$ is the width. Given $A=x^{3}+3x^{2}-18x$ and $w = x + 6$, we need to find $l=\frac{A}{w}=\frac{x^{3}+3x^{2}-18x}{x + 6}$.

Step2: Factor the numerator

Factor out $x$ from the numerator: $x^{3}+3x^{2}-18x=x(x^{2}+3x - 18)$. Then factor the quadratic $x^{2}+3x - 18=(x + 6)(x-3)$. So the numerator becomes $x(x + 6)(x - 3)$.

Step3: Simplify the fraction

We have $\frac{x(x + 6)(x - 3)}{x + 6}$. Cancel out the common factor $(x + 6)$ in the numerator and the denominator.

Answer:

$x(x - 3)=x^{2}-3x$