$A=\dfrac{1}{2}(x^2+y^2)^2-2x^2y^2$
$=\dfrac{1}{2}(x^4+y^4+2x^2y^2)-2x^2y^2$
$=\dfrac{1}{2}(x^4+y^4-2x^2y^2+4x^2y^2)-2x^2y^2$
$=\dfrac{1}{2}[(x^2-y^2)^2+4x^2y^2]-2x^2y^2$
$=\dfrac{1}{2}(x^2-y^2)+2x^2y^2-2x^2y^2$
$=\dfrac{1}{2}(x^2-y^2)^2$
$=\dfrac{1}{2}4^2$
$=8$