Simplify ((3x-15)/(x^2-25))÷((12(x+1))/(x^2+3x-10))

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

Factor ${\displaystyle 3}$ out of ${\displaystyle -15}$.

Factor ${\displaystyle 3}$ out of ${\displaystyle 3x+3\cdot -5}$.

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 ${\displaystyle 3}$.

Multiply ${\displaystyle \frac{x-5}{(x+5)(x-5)}}$ by ${\displaystyle \frac{x^2+3x-10}{4(x+1)}}$.

Cancel the common factor of ${\displaystyle x-5}$.

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 ${\displaystyle -10}$ and whose sum is ${\displaystyle 3}$.

Write the factored form using these integers.

Cancel the common factor.

Rewrite the expression.