Simplify (d-4)/(d^2+2d-8)-(d+2)/(d^2-16)

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 ${\displaystyle 2}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 16}$ as ${\displaystyle 4^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=d}$ and ${\displaystyle b=4}$.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify each term.

Subtract ${\displaystyle 4d}$ from ${\displaystyle -4d}$.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify each term.

Subtract ${\displaystyle 2d}$ from ${\displaystyle 2d}$.

Add ${\displaystyle -d^2}$ and ${\displaystyle 0}$.

Factor ${\displaystyle 4}$ out of ${\displaystyle -8d}$.

Factor ${\displaystyle 4}$ out of ${\displaystyle 20}$.

Factor ${\displaystyle 4}$ out of ${\displaystyle 4(-2d)+4\left(5\right)}$.

Rewrite ${\displaystyle -(2d-5)}$ as ${\displaystyle -1(2d-5)}$.

Move the negative in front of the fraction.