Simplify ((x^2-4)/(x-3))÷((x^2+7x-18)/(x+9))

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

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 ${\displaystyle b=2}$.

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 ${\displaystyle -18}$ and whose sum is ${\displaystyle 7}$.

Write the factored form using these integers.

Cancel the common factor of ${\displaystyle x-2}$.

Multiply ${\displaystyle \frac{x+2}{x-3}}$ by ${\displaystyle \frac{x+9}{x+9}}$.

Cancel the common factor of ${\displaystyle x+9}$.