Question

find the quotient. write your answer in simplest form.\\(
\frac{4x^2 - 9}{6x^2 + 13x + 6} \div \frac{4x^2 - 1}{6x^2 + x - 2}
\\)
(1 point)
\\(\circ \frac{2x + 3}{3x + 2}\\)\\(\circ \frac{2x - 3}{2x + 1}\\)\\(\circ \frac{2x - 3}{2x - 1}\\)\\(\circ \frac{3x + 2}{2x + 1}\\)

Explanation:

Step1: Rewrite division as multiplication

$\frac{4x^2 - 9}{6x^2 + 13x + 6} \times \frac{6x^2 + x - 2}{4x^2 - 1}$

Step2: Factor all polynomials

• $4x^2-9=(2x-3)(2x+3)$
• $6x^2+13x+6=(3x+2)(2x+3)$
• $6x^2+x-2=(3x+2)(2x-1)$
• $4x^2-1=(2x-1)(2x+1)$

Substitute:
$\frac{(2x-3)(2x+3)}{(3x+2)(2x+3)} \times \frac{(3x+2)(2x-1)}{(2x-1)(2x+1)}$

Step3: Cancel common factors

Cancel $(2x+3)$, $(3x+2)$, $(2x-1)$:
$\frac{2x-3}{1} \times \frac{1}{2x+1}$

Step4: Multiply remaining terms

$\frac{2x-3}{2x+1}$

Answer:

$\boldsymbol{\frac{2x-3}{2x+1}}$ (matches option B. $\frac{2x-3}{2x+1}$)