Step 1 :

To factor $ 18x^{3}+63x^{2}-8x-28 $ we can use factoring by grouping:

Group $ \color{blue}{ 18x^{3} }$ with $ \color{blue}{ 63x^{2} }$ and $ \color{red}{ -8x }$ with $ \color{red}{ -28 }$ then factor each group.

$$ \begin{aligned} 18x^{3}+63x^{2}-8x-28 = ( \color{blue}{ 18x^{3}+63x^{2} } ) + ( \color{red}{ -8x-28 }) &= \\ &= \color{blue}{ 9x^{2}( 2x+7 )} + \color{red}{ -4( 2x+7 ) } = \\ &= (9x^{2}-4)(2x+7) \end{aligned} $$

Step 2 :

Rewrite $ 9x^{2}-4 $ as:

$$ 9x^{2}-4 = (3x)^2 - (2)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = 3x $ and $ II = 2 $ , we have:

$$ 9x^{2}-4 = (3x)^2 - (2)^2 = ( 3x-2 ) ( 3x+2 ) $$

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