Simplify ((x^2+9x+18)/(x-5))÷((x+3)/(x^2-25))

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 ${\displaystyle 18}$ and whose sum is ${\displaystyle 9}$.

Write the factored form using these integers.

Cancel the common factor of ${\displaystyle x+3}$.

Multiply ${\displaystyle \frac{x+6}{x-5}}$ by ${\displaystyle x^2-25}$.

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.

Divide ${\displaystyle (x+6)(x+5)}$ by ${\displaystyle 1}$.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify each term.

Add ${\displaystyle 5x}$ and ${\displaystyle 6x}$.