Factor polynomial $$ s^{3}+3s^{2}-s-3 $$

$$ s^{3}+3s^{2}-s-3 = ( s-1 )( s+1 )( s+3 ) $$

Step 1 :

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

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

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

Step 2 :

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

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

Now we can apply the difference of squares formula.

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

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

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

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