Step 1 :

To factor $ x^{6}+2x^{4}-16x^{2}-32 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{6} }$ with $ \color{blue}{ 2x^{4} }$ and $ \color{red}{ -16x^{2} }$ with $ \color{red}{ -32 }$ then factor each group.

$$ \begin{aligned} x^{6}+2x^{4}-16x^{2}-32 = ( \color{blue}{ x^{6}+2x^{4} } ) + ( \color{red}{ -16x^{2}-32 }) &= \\ &= \color{blue}{ x^{4}( x^{2}+2 )} + \color{red}{ -16( x^{2}+2 ) } = \\ &= (x^{4}-16)(x^{2}+2) \end{aligned} $$

Step 2 :

Rewrite $ x^{4}-16 $ as:

$$ x^{4}-16 = (x^{2})^2 - (4)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = x^{2} $ and $ II = 4 $ , we have:

$$ x^{4}-16 = (x^{2})^2 - (4)^2 = ( x^{2}-4 ) ( x^{2}+4 ) $$

Step 3 :

Rewrite $ x^{2}-4 $ as:

$$ x^{2}-4 = (x)^2 - (2)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = x $ and $ II = 2 $ , we have:

$$ x^{2}-4 = (x)^2 - (2)^2 = ( x-2 ) ( x+2 ) $$

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