Simplify (x^2+7x+12)/(x^2-4)*(x+2)/(x^2-16)

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 ${\displaystyle 12}$ and whose sum is ${\displaystyle 7}$.

Write the factored form using these integers.

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 ${\displaystyle b=2}$.

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 of ${\displaystyle x+4}$.

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

Multiply ${\displaystyle \frac{x+3}{x-2}}$ by ${\displaystyle \frac{1}{x-4}}$.