Factor polynomial $$ 4cx^{2}-12x^{2}-c+3 $$

$$ 4cx^{2}-12x^{2}-c+3 = (2x+1)(2x-1)(c-3) $$

Step 1 :

To factor $ 4cx^2-12x^2-c+3 $ we can use factoring by grouping.

Group $ \color{blue}{ 4cx^2 }$ with $ \color{blue}{ -12x^2 }$ and $ \color{red}{ -c }$ with $ \color{red}{ 3 }$ then factor each group.

$$ \begin{aligned} 4cx^2-12x^2-c+3 &= ( \color{blue}{ 4cx^2-12x^2 } ) + ( \color{red}{ -c+3 }) = \\ &= \color{blue}{ 4x^2( c-3 )} \color{red}{ -1( c-3 ) } = \\ &= (4x^2-1)(c-3) \end{aligned} $$

Step 2 :

Rewrite $ 4x^2-1 $ as:

$$ \color{blue}{ 4x^2-1 = (2x)^2 - (1)^2 } $$

Now we can apply the difference of squares formula.

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

After putting $ I = 2x $ and $ II = 1 $ , we have:

$$ 4x^2-1 = (2x)^2 - (1)^2 = ( 2x-1 ) ( 2x+1 ) $$

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