$\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$ 
I need to evaluate $$\displaystyle\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$$

I think I'll need $\displaystyle\int^{\infty}_{-\infty} e^{-x^2} dx\,=\sqrt{\pi}$ . But I'm not able to apply it properly.

OK, the problem seems to be completing the square. What one should aim for is to collect the mixed term $xy$ into a squared term. Here I do it with the $x^2$-term:
$$
2x^2+2xy+2y^2=2\bigl(x+\tfrac{1}{2}y\bigr)^2+\frac{3}{2}y^2.
$$
Once that is done, you can change variables (for example $u=\sqrt{2}(x+y/2)$ and $v=\sqrt{3/2}y$. Can you proceed from here?
