Factorize
${(2 x + 3 y)}^{3} - {(2 x - 3 y)}^{3}$

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Solution

$(a + b)^{3} - (a - b)^{3}$
$= a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2} - (a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2})$
$= 2 b^{3} + 6 a^{2} b$
$= 2 b (b^{2} + 3 a^{2})$
$\begin{matrix} Thus & (2 x + 3 y)^{3} - (2 x - 3 y)^{3} = 3 y ((3 y)^{2} + 3 \cdot (2 x)^{2}) = 3 y (9 y^{2} + 12 x^{2}) \end{matrix}$
$= 9 y (3 y^{2} + 4 x^{2})$

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