Simplify (x-5)/(x^4-2x^3)*(x^2-4)/(x^2-3x-10)

Factor ${\displaystyle x^3}$ out of ${\displaystyle x^4}$.

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

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

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

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 ${\displaystyle b=2}$.

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 ${\displaystyle -10}$ and whose sum is ${\displaystyle -3}$.

Write the factored form using these integers.

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

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

Multiply ${\displaystyle \frac{1}{x^3}}$ by ${\displaystyle \frac{x+2}{x+2}}$.

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