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

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 ${\displaystyle -3}$ and whose sum is ${\displaystyle 2}$.

Write the factored form using these integers.

Cancel the common factor.

Rewrite the expression.

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

Raise ${\displaystyle x-1}$ to the power of ${\displaystyle 1}$.

Raise ${\displaystyle x-1}$ to the power of ${\displaystyle 1}$.

Use the power rule ${\displaystyle a^m{a}^n=a^{m+n}}$ to combine exponents.

Add ${\displaystyle 1}$ and ${\displaystyle 1}$.

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

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

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 ${\displaystyle b=x}$.

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

Cancel the common factor of ${\displaystyle 1-x}$ and ${\displaystyle {(x-1)}^2}$.