Step 1 :

To factor $ 25x^{3}+25x^{2}-x-1 $ we can use factoring by grouping:

Group $ \color{blue}{ 25x^{3} }$ with $ \color{blue}{ 25x^{2} }$ and $ \color{red}{ -x }$ with $ \color{red}{ -1 }$ then factor each group.

$$ \begin{aligned} 25x^{3}+25x^{2}-x-1 = ( \color{blue}{ 25x^{3}+25x^{2} } ) + ( \color{red}{ -x-1 }) &= \\ &= \color{blue}{ 25x^{2}( x+1 )} + \color{red}{ -1( x+1 ) } = \\ &= (25x^{2}-1)(x+1) \end{aligned} $$

Step 2 :

Rewrite $ 25x^{2}-1 $ as:

$$ 25x^{2}-1 = (5x)^2 - (1)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = 5x $ and $ II = 1 $ , we have:

$$ 25x^{2}-1 = (5x)^2 - (1)^2 = ( 5x-1 ) ( 5x+1 ) $$

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