The function is $f(x, y, z)=x^2+y^2-z$ subjected to $z=2x^2y^2+1$ .
My first step was to define $g(x, y, z)=z-2x^2y^2=1$
So $\nabla f = 2xi+2yj-k$, and $\lambda \nabla g = -4xy^2\lambda i -4x^2y\lambda j +\lambda k$
So I had the 3 equations
$$2x=-4xy^2$$ $$2y=-4x^2y$$ $$-1=\lambda$$
Substituting lambda we get
$2x=4xy^2$
$2y=4x^2y$
The solutions to this system of equations is $(0, 0)$ and $(\pm\dfrac{\sqrt2}{2}, \pm\dfrac{\sqrt2}{2})$
Is this the correct way to do it? It makes me a little nervous that I don't have a $z$