Simplify ((x^2+4x-12)/(x^2-25))÷((x+6)/(x^2-5x))

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 ${\displaystyle 4}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 25}$ as ${\displaystyle 5^2}$.

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 ${\displaystyle b=5}$.

Cancel the common factor of ${\displaystyle x+6}$.

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

Factor ${\displaystyle x}$ out of ${\displaystyle x^2}$.

Factor ${\displaystyle x}$ out of ${\displaystyle -5x}$.

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

Cancel the common factor.

Rewrite the expression.