Simplify (x^2+3x-4)/(3x^2-48)*(x^3-49x)/(x^2+6x-7)

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.

Factor ${\displaystyle 3}$ out of ${\displaystyle 3x^2-48}$.

Rewrite ${\displaystyle 16}$ as ${\displaystyle 4^2}$.

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 ${\displaystyle b=4}$.

Factor ${\displaystyle x}$ out of ${\displaystyle x^3-49x}$.

Rewrite ${\displaystyle 49}$ as ${\displaystyle 7^2}$.

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 ${\displaystyle b=7}$.

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

Write the factored form using these integers.

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

Multiply ${\displaystyle \frac{x+4}{3(x+4)(x-4)}}$ by ${\displaystyle \frac{x(x+7)(x-7)}{x+7}}$.

Cancel the common factor of ${\displaystyle x+4}$.

Cancel the common factor of ${\displaystyle x+7}$.