Simplify (1/(3x^2-3))/(5/(x+1)-(x+4)/(x^2-3x-4))

Factor ${\displaystyle 3}$ out of ${\displaystyle 3x^2-3}$.

Rewrite ${\displaystyle 1}$ as ${\displaystyle 1^2}$.

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 ${\displaystyle b=1}$.

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 ${\displaystyle -4}$ and whose sum is ${\displaystyle -3}$.

Write the factored form using these integers.

To write ${\displaystyle \frac{5}{x+1}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{x-4}{x-4}}$.

Write each expression with a common denominator of ${\displaystyle (x+1)(x-4)}$, by multiplying each by an appropriate factor of ${\displaystyle 1}$.

Combine the numerators over the common denominator.

Rewrite ${\displaystyle \frac{5(x-4)-(x+4)}{(x+1)(x-4)}}$ in a factored form.

Factor ${\displaystyle x+1}$ out of ${\displaystyle 3(x+1)(x-1)}$.

Cancel the common factor.

Rewrite the expression.