Simplify (4/(-x-1)*(x^2-11x+30)/(5-x))÷((x^2-7x+6)/(1-x^2))

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 ${\displaystyle 30}$ and whose sum is ${\displaystyle -11}$.

Write the factored form using these integers.

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

Rewrite using the commutative property of multiplication.

Apply the distributive property.

Combine ${\displaystyle x}$ and ${\displaystyle \frac{4}{-x-1}}$.

Multiply ${\displaystyle -6}$ by ${\displaystyle -1}$.

Combine ${\displaystyle 6}$ and ${\displaystyle \frac{4}{-x-1}}$.

Multiply ${\displaystyle 6}$ by ${\displaystyle 4}$.

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 ${\displaystyle a=1}$ and ${\displaystyle b=x}$.

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 ${\displaystyle 6}$ and whose sum is ${\displaystyle -7}$.

Write the factored form using these integers.

Combine the numerators over the common denominator.

Multiply ${\displaystyle 4}$ by ${\displaystyle -1}$.

Factor ${\displaystyle 4}$ out of ${\displaystyle -4x+24}$.

Cancel the common factor of ${\displaystyle 1-x}$ and ${\displaystyle x-1}$.

Simplify the expression.

Multiply ${\displaystyle \frac{1+x}{x-6}}$ by ${\displaystyle \frac{4(-x+6)}{-x-1}}$.

Cancel the common factor of ${\displaystyle 1+x}$ and ${\displaystyle -x-1}$.

Cancel the common factor of ${\displaystyle -x+6}$ and ${\displaystyle x-6}$.

Multiply.