$\begin{cases} y\geq 4x^4+4yx^2+\frac{1}{2}\\y^2-2x^2\leq \frac{1}{4}\end{cases}$
In desmos, it is clearly visible that this is a limitation task (and the task is in the limitations section). I have an assumption that
$4x^4+4yx^2+\frac{1}{2}-y\geq0$, if $y^2-2x^2\leq \frac{1}{4}$
but this is just a guess. I tried to distribute the sum of the squares, but I couldn't. I tried to replace the second inequality with the first one, but nothing worked. I will be glad if you can help!