I want to find when the gradient of the following function vanishes
$$ f(x,y) = \ln(1+x^2 + 2y^2 ) + x^2 y^2 + y.$$ 
Nevertheless I have trouble to solve one part. 
I have to find when this polynomial
$$ 2x + 2y^2 + 2 x^2 y^2 + 4y^4 $$ is equal to $0$. I tried to use the method of completing the squares (of Gauss I assume) but in vain.
I have the intuition that it never goes through zero, but I have no proof.