Expanding (x+y)^2, we get:

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

Now, substituting this back into the original expression, we have:

(x+y)^2 - x^4 - y^4 + 2x^2y^2
= (x^2 + 2xy + y^2) - x^4 - y^4 + 2x^2y^2
= x^2 + 2xy + y^2 - x^4 - y^4 + 2x^2y^2

Now, let's simplify further by expanding and combining like terms:

= x^2 + 2xy + y^2 - x^4 - y^4 + 2x^2y^2
= x^2 - x^4 + 2xy + 2x^2y^2 + y^2 - y^4
= -x^4 + x^2 + 2x^2y^2 + 2xy + y^2 - y^4

This is the simplified expression for (x+y)^2 - x^4 - y^4 + 2x^2y^2.