Simplify ((x+1)/(x+1)*(x^2-1)/(4x^2-40x))÷((x^2+9x-10)/(x^2-100))

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

Multiply ${\displaystyle \frac{x^2-1}{4x^2-40x}}$ by ${\displaystyle 1}$.

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 ${\displaystyle b=1}$.

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

Factor ${\displaystyle 4x}$ out of ${\displaystyle -40x}$.

Factor ${\displaystyle 4x}$ out of ${\displaystyle 4x\left(x\right)+4x(-10)}$.

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 ${\displaystyle b=10}$.

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 ${\displaystyle 9}$.

Write the factored form using these integers.

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

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

Multiply ${\displaystyle \frac{x+1}{4x}}$ by ${\displaystyle \frac{x+10}{x+10}}$.

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