Factor polynomial $$ 2p^{3}q-2p^{3}+p^{2}q-p^{2} $$

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

Step 1 :

Factor out common factor $ \color{blue}{ p^2 } $:

$$ 2p^3q-2p^3+p^2q-p^2 = p^2 ( 2pq-2p+q-1 ) $$

Step 2 :

To factor $ 2pq-2p+q-1 $ we can use factoring by grouping.

Group $ \color{blue}{ 2pq }$ with $ \color{blue}{ -2p }$ and $ \color{red}{ q }$ with $ \color{red}{ -1 }$ then factor each group.

$$ \begin{aligned} 2pq-2p+q-1 &= ( \color{blue}{ 2pq-2p } ) + ( \color{red}{ q-1 }) = \\ &= \color{blue}{ 2p( q-1 )} + \color{red}{ 1( q-1 ) } = \\ &= (2p+1)(q-1) \end{aligned} $$

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