Factor polynomial $$ x^{6}+5x^{5}-25x^{4}-125x^{3} $$

$$ x^{6}+5x^{5}-25x^{4}-125x^{3} = x^{3}( x-5 )( x+5 )^{ 2 } $$

Step 1 :

After factoring out $ x^{3} $ we have:

$$ x^{6}+5x^{5}-25x^{4}-125x^{3} = x^{3} ( x^{3}+5x^{2}-25x-125 ) $$

Step 2 :

To factor $ x^{3}+5x^{2}-25x-125 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{3} }$ with $ \color{blue}{ 5x^{2} }$ and $ \color{red}{ -25x }$ with $ \color{red}{ -125 }$ then factor each group.

$$ \begin{aligned} x^{3}+5x^{2}-25x-125 = ( \color{blue}{ x^{3}+5x^{2} } ) + ( \color{red}{ -25x-125 }) &= \\ &= \color{blue}{ x^{2}( x+5 )} + \color{red}{ -25( x+5 ) } = \\ &= (x^{2}-25)(x+5) \end{aligned} $$

Step 3 :

Rewrite $ x^{2}-25 $ as:

$$ x^{2}-25 = (x)^2 - (5)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = x $ and $ II = 5 $ , we have:

$$ x^{2}-25 = (x)^2 - (5)^2 = ( x-5 ) ( x+5 ) $$

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