Simplify ((x^2+2x-24)/(x^2-9))÷((x^2-16)/(x+3))

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 ${\displaystyle 2}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 9}$ as ${\displaystyle 3^2}$.

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 ${\displaystyle b=3}$.

Cancel the common factor of ${\displaystyle x+3}$.

Multiply ${\displaystyle \frac{(x-4)(x+6)}{x-3}}$ by ${\displaystyle \frac{1}{x^2-16}}$.

Rewrite ${\displaystyle 16}$ as ${\displaystyle 4^2}$.

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 ${\displaystyle b=4}$.

Cancel the common factor.

Rewrite the expression.