Step 1 :

To factor $ 4a^{3}-a^{2}-4a+1 $ we can use factoring by grouping:

Group $ \color{blue}{ 4x^{3} }$ with $ \color{blue}{ -x^{2} }$ and $ \color{red}{ -4x }$ with $ \color{red}{ 1 }$ then factor each group.

$$ \begin{aligned} 4a^{3}-a^{2}-4a+1 = ( \color{blue}{ 4x^{3}-x^{2} } ) + ( \color{red}{ -4x+1 }) &= \\ &= \color{blue}{ x^{2}( 4x-1 )} + \color{red}{ -1( 4x-1 ) } = \\ &= (x^{2}-1)(4x-1) \end{aligned} $$

Step 2 :

Rewrite $ a^{2}-1 $ as:

$$ a^{2}-1 = (a)^2 - (1)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = a $ and $ II = 1 $ , we have:

$$ a^{2}-1 = (a)^2 - (1)^2 = ( a-1 ) ( a+1 ) $$

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