Simplify ((3x)/(x^2+10x+24))÷((x-6)/(x^2-36))

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 ${\displaystyle 24}$ and whose sum is ${\displaystyle 10}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 36}$ as ${\displaystyle 6^2}$.

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 ${\displaystyle b=6}$.

Cancel the common factor of ${\displaystyle x+6}$.

Multiply ${\displaystyle \frac{3x}{x+4}}$ by ${\displaystyle \frac{x-6}{x-6}}$.

Cancel the common factor of ${\displaystyle x-6}$.