Factor polynomial $$ 3v+3-27u^{2}-27u^{2}v $$

$$ 3v+3-27u^{2}-27u^{2}v = 3(1+3u)(1-3u)(v+1) $$

Step 1 :

Factor out common factor $ \color{blue}{ 3 } $:

$$ 3v+3-27u^2-27u^2v = 3 ( v+1-9u^2-9u^2v ) $$

Step 2 :

To factor $ v+1-9u^2-9u^2v $ we can use factoring by grouping.

Group $ \color{blue}{ v }$ with $ \color{blue}{ 1 }$ and $ \color{red}{ -9u^2 }$ with $ \color{red}{ -9u^2v }$ then factor each group.

$$ \begin{aligned} v+1-9u^2-9u^2v &= ( \color{blue}{ v+1 } ) + ( \color{red}{ -9u^2-9u^2v }) = \\ &= \color{blue}{ 1( v+1 )} \color{red}{ -9u^2( 1+v ) } = \\ &= (1-9u^2)(v+1) \end{aligned} $$

Step 3 :

Rewrite $ 1-9u^2 $ as:

$$ \color{blue}{ 1-9u^2 = (1)^2 - (3u)^2 } $$

Now we can apply the difference of squares formula.

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

After putting $ I = 1 $ and $ II = 3u $ , we have:

$$ 1-9u^2 = (1)^2 - (3u)^2 = ( 1-3u ) ( 1+3u ) $$

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