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

Oct 8, 2016

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

$4 {x}^{2} + 8 x y + 4 {y}^{2} = {\left(2 x + 2 y\right)}^{2}$

#### Explanation:

$4 {x}^{2} + 8 x y + 4 {y}^{2} = 4 {x}^{2} + 4 x y + 4 x y + 4 {y}^{2}$

$= \left[4 {x}^{2} + 4 x y\right] + \left[4 x y + 4 {y}^{2}\right]$

$= 2 x \left(2 x + 2 y\right) + 2 y \left(2 x + 2 y\right)$

$= \left(2 x + 2 y\right) \left(2 x + 2 y\right)$

$= {\left(2 x + 2 y\right)}^{2}$