Question

the length of a rectangle is ( 3x + 2 ). its area is ( 21x^{2}-x - 10 ). to find an expression that represents the width of the rectangle, perform the following operation(s) and simplify completely. ( \frac{21x^{2}-x - 10}{3x + 2} ) (1 point)

Explanation:

Step1: Factor the numerator

For the quadratic expression \(21x^{2}-x - 10\), we use the formula \(ax^{2}+bx + c\). We need to find two numbers \(m\) and \(n\) such that \(m\times n=21\times(- 10)=-210\) and \(m + n=-1\). The numbers are \(m = 14\) and \(n=-15\).

$$
LATEXBLOCK0
$$

Step2: Simplify the fraction

We have \(\frac{21x^{2}-x - 10}{3x + 2}\), substituting the factored form of the numerator \((3x + 2)(7x-5)\) into the fraction:

$$ \frac{(3x + 2)(7x-5)}{3x + 2} $$

Cancel out the common factor \((3x + 2)\) (assuming \(x
eq-\frac{2}{3}\))

Answer:

\(7x-5\)