Simplify $x^{4} + 2 x^{3} y - 2 x y^{3} - y^{4}$

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Solution

We have,

$x^{4} + 2 x^{3} y - 2 x y^{3} - y^{4}$

$= x^{4} - y^{4} + 2 x^{3} y - 2 x y^{3}$

$= {(x^{2})}^{2} - {(y^{2})}^{2} + 2 x y (x^{2} - y^{2})$

$= (x^{2} - y^{2}) (x^{2} + y^{2}) + 2 x y (x^{2} - y^{2})$

$= (x^{2} - y^{2}) (x^{2} + y^{2} + 2 x y)$

$= (x^{2} - y^{2}) {(x + y)}^{2}$

$= (x + y) (x - y) {(x + y)}^{2}$

$= {(x + y)}^{3} (x - y)$
Hence, this is the answer.

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