Simplify: $(x + y)^{3} - (x - y)^{3} - 6 y (x^{2} - y^{2})$ .

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Solution

$= (x + y)^{3} - (x - y)^{3} - 6 y (x^{2} - y^{2})$
$= (x + y)^{3} - (x - y)^{3} - 6 y (x - y) (x + y)$
$= (x + y)^{3} - (x - y) ((x - y)^{2} + 6 y (x + y))$
$= (x + y)^{3} - (x - y) (x^{2} + y^{2} - 2 x y + 6 x y + 6 y^{2})$
$= (x + y)^{3} - (x - y) (x^{2} + 7 y^{2} + 4 x y)$
$= x^{3} + y^{3} + 3 x^{2} y + 3 x y^{2} - x^{3} - 7 x y^{2} - 4 x^{2} y + x^{2} y + 7 y^{3} + 4 x y^{2}$
$= 8 y^{3}$

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