Simplify ((x^2-6x+8)/(3x-12))÷((x^2-4)/(x^2+5x+6))

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 ${\displaystyle 8}$ and whose sum is ${\displaystyle -6}$.

Write the factored form using these integers.

Factor ${\displaystyle 3}$ out of ${\displaystyle 3x-12}$.

Cancel the common factor of ${\displaystyle x-4}$.

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 ${\displaystyle 6}$ and whose sum is ${\displaystyle 5}$.

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 ${\displaystyle b=2}$.

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

Multiply ${\displaystyle \frac{1}{3}}$ by ${\displaystyle \frac{(x+2)(x+3)}{x+2}}$.

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