Question

the area, a, of a rectangle is 120x² + 78x - 90, and the length, l, of the rectangle is 12x + 15. which of the following gives the width, w, of the rectangle?
9x + 4
10x - 10
10x - 6
8x - 6

Explanation:

Step1: Recall area formula

The area of a rectangle is $A = l\times w$, so $w=\frac{A}{l}$. Here, $A = 120x^{2}+78x - 90$ and $l=12x + 15$.

Step2: Factor expressions

Factor out the greatest - common factor from $A$ and $l$. For $A = 120x^{2}+78x - 90$, the GCF is 6, so $A=6(20x^{2}+13x - 15)$. For $l = 12x + 15$, the GCF is 3, so $l = 3(4x+5)$.

Step3: Perform polynomial division

We need to divide $\frac{6(20x^{2}+13x - 15)}{3(4x + 5)}$. First, simplify the constant part: $\frac{6}{3}=2$. Then, perform the polynomial long - division or factor $20x^{2}+13x - 15=(4x + 5)(5x-3)$. So, $\frac{6(20x^{2}+13x - 15)}{3(4x + 5)}=\frac{6(4x + 5)(5x-3)}{3(4x + 5)}$.

Step4: Simplify the result

Cancel out the common factor $(4x + 5)$ and simplify $\frac{6}{3}=2$. We get $2(5x-3)=10x-6$.

Answer:

$10x - 6$