Step 1 :

To factor $ r^{3}-5r^{2}-100r+500 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{3} }$ with $ \color{blue}{ -5x^{2} }$ and $ \color{red}{ -100x }$ with $ \color{red}{ 500 }$ then factor each group.

$$ \begin{aligned} r^{3}-5r^{2}-100r+500 = ( \color{blue}{ x^{3}-5x^{2} } ) + ( \color{red}{ -100x+500 }) &= \\ &= \color{blue}{ x^{2}( x-5 )} + \color{red}{ -100( x-5 ) } = \\ &= (x^{2}-100)(x-5) \end{aligned} $$

Step 2 :

Rewrite $ r^{2}-100 $ as:

$$ r^{2}-100 = (r)^2 - (10)^2 $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = r $ and $ II = 10 $ , we have:

$$ r^{2}-100 = (r)^2 - (10)^2 = ( r-10 ) ( r+10 ) $$

Polynomial Factoring Calculator