The value of $(2 x - 3 y)^{3} + (x + 3 y)^{3} + 3 (2 x - 3 y)^{2} (x + 3 y) + 3 (2 x - 3 y) (x + 3 y)^{2}$ is

27 x^{3}

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27 x^{3} + 9 y^{3}

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Solution

The correct option is A $27 x^{3}$
Substitute $2 x - 3 y = a$ and $(x + 3 y) = b$
$(2 x - 3 y)^{3} + (x + 3 y)^{3} + 3 (2 x - 3 y)^{2} (x + 3 y) + 3 (2 x - 3 y) (x + 3 y)^{2}$
$= a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
Now, on using the identity:
$(a + b)^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
$= (a + b)^{3}$
$= (2 x - 3 y + x + 3 y)^{3}$
$= (3 x)^{3}$
$= 27 x^{3}$

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