Question

perform the following operation and express in simplest form.
$\frac{x^{2}-49}{x^{4}+7x^{3}} div \frac{x^{2}-x-42}{3x}$

Explanation:

Step1: Rewrite division as multiplication

$\frac{x^2 - 49}{x^4 + 7x^3} \times \frac{3x}{x^2 - x - 42}$

Step2: Factor all polynomials

$\frac{(x-7)(x+7)}{x^3(x+7)} \times \frac{3x}{(x-7)(x+6)}$

Step3: Cancel common factors

Cancel $(x+7)$, $(x-7)$, and one $x$:
$\frac{1}{x^2} \times \frac{3}{x+6}$

Step4: Multiply remaining terms

$\frac{3}{x^2(x+6)}$

Answer:

$\frac{3}{x^3 + 6x^2}$