Step 1 :

To factor $ x^{10}+x^{6}+x^{4}+1 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{10} }$ with $ \color{blue}{ x^{6} }$ and $ \color{red}{ x^{4} }$ with $ \color{red}{ 1 }$ then factor each group.

$$ \begin{aligned} x^{10}+x^{6}+x^{4}+1 = ( \color{blue}{ x^{10}+x^{6} } ) + ( \color{red}{ x^{4}+1 }) &= \\ &= \color{blue}{ x^{6}( x^{4}+1 )} + \color{red}{ 1( x^{4}+1 ) } = \\ &= (x^{6}+1)(x^{4}+1) \end{aligned} $$

Step 2 :

To factor $ x^{6}+1 $ we can use sum of cubes formula:

$$ I^3 - II^3 = (I + II)(I^2 - I \cdot II + II^2) $$

After putting $ I = x^{2} $ and $ II = 1 $ , we have:

$$ x^{6}+1 = ( x^{2}+1 ) ( x^{4}-x^{2}+1 ) $$

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