Simplify ((24-6x)/(x^2-10x+24)*(x^2-8x+12)/(10-x))÷((x^2+8x-20)/(100x-x^3))

Factor ${\displaystyle 6}$ out of ${\displaystyle 24}$.

Factor ${\displaystyle 6}$ out of ${\displaystyle -6x}$.

Factor ${\displaystyle 6}$ out of ${\displaystyle 6\left(4\right)+6(-x)}$.

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 ${\displaystyle 24}$ and whose sum is ${\displaystyle -10}$.

Write the factored form using these integers.

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 ${\displaystyle 12}$ and whose sum is ${\displaystyle -8}$.

Write the factored form using these integers.

Cancel the common factor of ${\displaystyle x-6}$.

Multiply ${\displaystyle \frac{6(4-x)}{x-4}}$ by ${\displaystyle \frac{x-2}{10-x}}$.

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

Simplify the expression.

Factor ${\displaystyle x}$ out of ${\displaystyle 100x-x^3}$.

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 ${\displaystyle a=10}$ and ${\displaystyle b=x}$.

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 ${\displaystyle -20}$ and whose sum is ${\displaystyle 8}$.

Write the factored form using these integers.

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

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

Combine ${\displaystyle -6}$ and ${\displaystyle \frac{x(10+x)}{x+10}}$.

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