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I think you might be missing a dy somewhere. Are you?
If your problem is (2x^2 + y^2)dx + 2xy dy = 0, then this is equivalent to
(2x^2 + y^2) + 2xy dy/dx = 0, or
2xy y' + y^2 = -2x^2.
If you know the method of integrating factors, this problem is now easy.
The idea is that you might recognize the Left Hand Side as
the derivative of the product [x*y^2] with respect to x.
Why? Because if you take the derivative of x*y^2 with respect to x, you use the product rule and you would get
y^2 + x*(2y)*y' (the y' comes from the chain rule since y is a function of x).
So now in our equation, if the Left Hand Side is the derivative of the product of x*y^2, then so is the Right Hand Side.
So integrate both sides to get that x*y^2 = -2(x^3)/3 + C.
Then solve for y if you want an explicit solution, or you can move things around so you get C on the Right Hand Side and everything else on the Left to get an implicit solution.