Bulgarian Spring Tournament

We have the inequalities $x^4\ge 0$, $5x^2(y^2+1)\ge 0$ and $4y^4+12y^3+y^2-12y+4=(2y-1{)}^2(y+2{)}^2\ge 0$. The sum of the left sides is 0 if and only if each of them is equal to 0. The first two lead to $x=0$, and the third to $y=-2$ or $y=\frac{1}{2}$. $\square$