Simplify ((20x-4x^2)/(2x^2+2x)*(x^2-100)/(x+10))÷((x^2-11x+10)/(x^2-1))

Factor ${\displaystyle 4x}$ out of ${\displaystyle 20x-4x^2}$.

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

Rewrite ${\displaystyle 100}$ as ${\displaystyle {10}^2}$.

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 ${\displaystyle b=10}$.

Cancel the common factor of ${\displaystyle 4}$ and ${\displaystyle 2}$.

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

Cancel the common factor of ${\displaystyle x+10}$.

Multiply ${\displaystyle \frac{2(5-x)}{x+1}}$ by ${\displaystyle x-10}$.

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 ${\displaystyle 10}$ and whose sum is ${\displaystyle -11}$.

Write the factored form using these integers.

Cancel the common factor of ${\displaystyle x-10}$.

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

Combine ${\displaystyle \frac{x-1}{x-1}}$ and ${\displaystyle 2}$.

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

Apply the distributive property.

Multiply.