Simplify ((x^2-49)/(x^4+7x^3))÷((x^2-x-42)/(3x))

Rewrite ${\displaystyle 49}$ as ${\displaystyle 7^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=x}$ and ${\displaystyle b=7}$.

Factor ${\displaystyle x^3}$ out of ${\displaystyle x^4+7x^3}$.

Cancel the common factor of ${\displaystyle x}$.

Multiply ${\displaystyle \frac{(x+7)(x-7)}{x^2(x+7)}}$ by ${\displaystyle \frac{3}{x^2-x-42}}$.

Cancel the common factor of ${\displaystyle x+7}$.

Consider the form ${\displaystyle x^2+bx+c}$. Find a pair of integers whose product is ${\displaystyle c}$ and whose sum is ${\displaystyle b}$. In this case, whose product is ${\displaystyle -42}$ and whose sum is ${\displaystyle -1}$.

Write the factored form using these integers.

Cancel the common factor.

Rewrite the expression.