Simplify ((x^3-8)/(x^2+3x-10))÷((8x-40)/(x^2-25))

Rewrite ${\displaystyle 8}$ as ${\displaystyle 2^3}$.

Since both terms are perfect cubes, factor using the difference of cubes formula, ${\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)}$ where ${\displaystyle a=x}$ and ${\displaystyle b=2}$.

Simplify.

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.

Rewrite the expression.

Rewrite ${\displaystyle 25}$ as ${\displaystyle 5^2}$.

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 ${\displaystyle b=5}$.

Factor ${\displaystyle 8}$ out of ${\displaystyle 8x-40}$.

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

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

Apply the distributive property.

Combine ${\displaystyle x^2}$ and ${\displaystyle \frac{1}{8}}$.

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

Combine ${\displaystyle x}$ and ${\displaystyle \frac{1}{4}}$.

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

Multiply ${\displaystyle \frac{x}{4}}$ by ${\displaystyle \frac{2}{2}}$.

Multiply ${\displaystyle 4}$ by ${\displaystyle 2}$.

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

Factor ${\displaystyle x}$ out of ${\displaystyle x\cdot 2}$.

Factor ${\displaystyle x}$ out of ${\displaystyle x\cdot x+x\left(2\right)}$.

Multiply ${\displaystyle \frac{1}{2}}$ by ${\displaystyle \frac{4}{4}}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle 4}$.

Apply the distributive property.

Multiply ${\displaystyle x}$ by ${\displaystyle x}$.

Move ${\displaystyle 2}$ to the left of ${\displaystyle x}$.