x and y error

It is given that $\space x = \sqrt{3}+2\sqrt{2}\space$ and $\space y = \sqrt{3}-2\sqrt{2}$

$\Rightarrow x + y = 2\sqrt{3} \space$ and $\space xy = (\sqrt{3})^2 - (2\sqrt{2})^2 = -5$

$(x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

$\Rightarrow x^4 + y^4 + 6x^2y^2 = (x+y)^4 - 4x^3y - 4xy^3 = (x+y)^4 - 4xy (x^2 + y^2)$

$\quad = (x+y)^4 - 4xy [(x+y)^2-2xy] = (2\sqrt{3})^4 - 4(-5)[(2\sqrt{3})^2 -2(-5)]$

$\quad = 144 + 20(22) = \boxed{584}$