Is $4x^2 + 8xy + 4y^2$ a perfect square?

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Answer 1 · 2016-10-08T00:39:25

Yes, as long as $2x+2y$ is an integer.

$4x^2+8xy+4y^2=(2x+2y)^2$

$4x^2+8xy+4y^2 = 4x^2+4xy+4xy+4y^2$

$=[4x^2+4xy]+[4xy+4y^2]$

$=2x(2x+2y)+2y(2x+2y)$

$=(2x+2y)(2x+2y)$

$=(2x+2y)^2$

Is 4x^2 + 8xy + 4y^24x^2 + 8xy + 4y^2 a perfect square?