Finding critical numbers of a product involving an exponential function

Ok, so the original function is $\displaystyle f(x,y)=(x^2+4y^2)e^{1-x^2-y^2}$ and the problem is to find the derivative with respect to x, respect to y, respect to xx, respect to yy, and respect to xy/yx and find the critical numbers as well as any relative extrema or saddle point if applicable.

I found that the derivative with respect to x is $\displaystyle f'x(x,y)=2xe^{1-x^2-y^2}(1-x^2-4y^2)$, the derivative with respect to y is $\displaystyle f'y(x,y)=2ye^{1-x^2-y^2}(4-x^2-4y^2)$, the derivative with respect to xx is $\displaystyle f'xx(x,y)=2e^{1-x^2-y^2}(1-5x^2+2x^4+8x^2y^2-4y^2)$, the derivative with respect to yy is $\displaystyle f'yy(x,y)=2e^{1-x^2-y^2}(4-x^2-20y^2+2x^2y^2+8y^4)$ and finally the derivative with respect to xy is $\displaystyle f'xy(x,y)=-4xye^{1-x^2-y^2}(5-x^2-4y^2)$.

Now, I tried setting $\displaystyle 2xe^{1-x^2-y^2}$ from $\displaystyle f'x$ equal to $\displaystyle 2ye^{1-x^2-y^2}$ from $\displaystyle f'y$ but that didn't work. I tried doing substitution for $\displaystyle 4-x^2-4y^2=0$ and $\displaystyle 1-x^2-4y^2=0$ but that ends up being 1=4 which obviously isn't true nor is it a CN...so how would I find the CN's for this problem? I'm sure I'm overlooking something simple, however, I can't find out what that simple thing is.