<s>[INST] Proving the maximum of $(x^2-y^2-1)^2+4x^2y^2$ occurs at $x=0$, $y=\pm 1$. I'm trying to get some practice using the Maximum Modulus theorem, and want to use it to conclude that the maximum of $(x^2-y^2-1)^2+4x^2y^2$ occurs at $x=0$, $y=\pm 1$, supposing $x^2+y^2\leq 1$. 
My thinking is I want to find some suitalbe complex function $f(z)$ such that $|f(z)|^2=(x^2-y^2-1)^2+4x^2y^2$, and then apply the Maximum Modulus principle, since I know the maximum will occur somewhere on the boundary $x^2+y^2=1$. However, I can't find such a function, so maybe I"m approaching it incorrectly.
I did manage to find that 
$$
z^2-1=(x+iy)^2-1=(x^2-y^2-1)+2xyi
$$
which looked somewhat close, but no cigar. How can this be done better? Thanks.
 [/INST] $x^2+y^2=1$
$(x^2-y^2-1)^2+4x^2y^2=(x^2+y^2)^2-2(x^2-y^2)+1=4-4x^2\leq 4$
$x=0,y=±1$
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