Step 1 :

After factoring out $ x $ we have:

$$ x^{4}+6x^{3}-36x^{2}-216x = x ( x^{3}+6x^{2}-36x-216 ) $$

Step 2 :

To factor $ x^{3}+6x^{2}-36x-216 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{3} }$ with $ \color{blue}{ 6x^{2} }$ and $ \color{red}{ -36x }$ with $ \color{red}{ -216 }$ then factor each group.

$$ \begin{aligned} x^{3}+6x^{2}-36x-216 = ( \color{blue}{ x^{3}+6x^{2} } ) + ( \color{red}{ -36x-216 }) &= \\ &= \color{blue}{ x^{2}( x+6 )} + \color{red}{ -36( x+6 ) } = \\ &= (x^{2}-36)(x+6) \end{aligned} $$

Step 3 :

Rewrite $ x^{2}-36 $ as:

$$ x^{2}-36 = (x)^2 - (6)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = x $ and $ II = 6 $ , we have:

$$ x^{2}-36 = (x)^2 - (6)^2 = ( x-6 ) ( x+6 ) $$

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