Simplify ((x^2-4)/(5x+30))÷((x^2+11x+18)/(3x+27))

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 ${\displaystyle b=2}$.

Factor ${\displaystyle 5}$ out of ${\displaystyle 5x+30}$.

Factor ${\displaystyle 3}$ out of ${\displaystyle 3x+27}$.

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 ${\displaystyle 11}$.

Write the factored form using these integers.

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

Multiply ${\displaystyle \frac{x-2}{5(x+6)}}$ by ${\displaystyle \frac{3(x+9)}{x+9}}$.

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

Move ${\displaystyle 3}$ to the left of ${\displaystyle x-2}$.