Factor polynomial $$ 48x^{3}+112x^{2}-3x-7 $$

$$ 48x^{3}+112x^{2}-3x-7 = ( 4x-1 )( 4x+1 )( 3x+7 ) $$

Step 1 :

To factor $ 48x^{3}+112x^{2}-3x-7 $ we can use factoring by grouping:

Group $ \color{blue}{ 48x^{3} }$ with $ \color{blue}{ 112x^{2} }$ and $ \color{red}{ -3x }$ with $ \color{red}{ -7 }$ then factor each group.

$$ \begin{aligned} 48x^{3}+112x^{2}-3x-7 = ( \color{blue}{ 48x^{3}+112x^{2} } ) + ( \color{red}{ -3x-7 }) &= \\ &= \color{blue}{ 16x^{2}( 3x+7 )} + \color{red}{ -1( 3x+7 ) } = \\ &= (16x^{2}-1)(3x+7) \end{aligned} $$

Step 2 :

Rewrite $ 16x^{2}-1 $ as:

$$ 16x^{2}-1 = (4x)^2 - (1)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = 4x $ and $ II = 1 $ , we have:

$$ 16x^{2}-1 = (4x)^2 - (1)^2 = ( 4x-1 ) ( 4x+1 ) $$

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