Simplify ((x^2+12x+35)/(x^2-49))÷((x+7)/(6x-42))

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 ${\displaystyle 35}$ and whose sum is ${\displaystyle 12}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 49}$ as ${\displaystyle 7^2}$.

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 ${\displaystyle b=7}$.

Cancel the common factor of ${\displaystyle x+7}$.

Multiply ${\displaystyle \frac{x+5}{(x+7)(x-7)}}$ by ${\displaystyle 6x-42}$.

Factor ${\displaystyle 6}$ out of ${\displaystyle 6x}$.

Factor ${\displaystyle 6}$ out of ${\displaystyle -42}$.

Factor ${\displaystyle 6}$ out of ${\displaystyle 6x+6\cdot -7}$.

Cancel the common factor.

Rewrite the expression.