Factor polynomial $$ -48x^{5}+128x^{4}-24x^{3}+64x^{2} $$

$$ -48x^{5}+128x^{4}-24x^{3}+64x^{2} = ( -8 )x^{2}( 2x^{2}+1 )( 3x-8 ) $$

Step 1 :

After factoring out $ -8x^{2} $ we have:

$$ -48x^{5}+128x^{4}-24x^{3}+64x^{2} = -8x^{2} ( 6x^{3}-16x^{2}+3x-8 ) $$

Step 2 :

To factor $ 6x^{3}-16x^{2}+3x-8 $ we can use factoring by grouping:

Group $ \color{blue}{ 6x^{3} }$ with $ \color{blue}{ -16x^{2} }$ and $ \color{red}{ 3x }$ with $ \color{red}{ -8 }$ then factor each group.

$$ \begin{aligned} 6x^{3}-16x^{2}+3x-8 = ( \color{blue}{ 6x^{3}-16x^{2} } ) + ( \color{red}{ 3x-8 }) &= \\ &= \color{blue}{ 2x^{2}( 3x-8 )} + \color{red}{ 1( 3x-8 ) } = \\ &= (2x^{2}+1)(3x-8) \end{aligned} $$

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