Proving global minimum by lower bound of 2-variable function $f(x,y)=x^4+2x^2y+y^2-4x^2-8x-8y$ I would like to prove that the following function $f :\mathbb{R}^2\to\mathbb{R}$ has a global minimum:

$f(x,y)=x^4+2x^2y+y^2-4x^2-8x-8y=(x^2+y)^2-4(x^2+2x+2y)$

$f$ has strict local minimum at
$f(1,3)=-20$
I think that what I need to show is that $-20$ is a lower bound of this function, and then conclude that's a global minimum, but I didn't manage to do so.
Please advise.
Thank you.

You can find the stationary points:
\begin{align}
\frac{\partial f}{\partial x}&=4x^3+4xy-8x-8 \\[6px]
\frac{\partial f}{\partial y}&=2x^2+2y-8
\end{align}
At a critical point $y=4-x^2$ and also
$$
x^3+x(4-x^2)-2x-2=0
$$
that is, $x=1$, that implies $y=3$.
Since clearly the function is upper unbounded on the line $y=0$, we just need to show it is lower bounded. Conjecturing that the stationary point is a minimum, we have $f(1,3)=-20$, we need to see whether $f(x,y)\ge-20$.
Now let's try completing the square in
$$
y^2+2(x^2-4)y+x^4-4x^2-8x+20
$$
Since $(x^2-4)^2=x^4-8x^2+16$, we have
$$
f(x,y)+20=(y+x^2-4)^2+4x^2-8x+4=(y+x^2-4)^2+4(x-1)^2
$$
which is everywhere nonnegative, so we proved that $f(x,y)\ge-20$.
