Find the value of $\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+xy+y^2)} \, dx\,dy$ 
Given that $\int_{-\infty}^\infty e^{-x^2} \, dx=\sqrt{\pi}$. Find the value of $$\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+xy+y^2)} \, dx\,dy$$

I don't understand how I find this double integral by using the given data. Please help.

$$
\begin{align}
\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+xy+y^2)}\,\mathrm{d}x\,\mathrm{d}y
&=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-((x+y/2)^2+3y^2/4)}\,\mathrm{d}x\,\mathrm{d}y\\
&=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(u^2+3y^2/4)}\,\mathrm{d}u\,\mathrm{d}y\\
&=\int_{-\infty}^\infty e^{-u^2}\,\mathrm{d}u
\int_{-\infty}^\infty e^{-3y^2/4}\,\mathrm{d}y\\
&=\int_{-\infty}^\infty e^{-u^2}\,\mathrm{d}u
\sqrt{\frac43}\int_{-\infty}^\infty e^{-v^2}\,\mathrm{d}v\\
&=\sqrt\pi\sqrt{\frac43}\sqrt\pi\\
&=\pi\sqrt{\frac43}\\
&=\frac{2\pi}{\sqrt3}
\end{align}
$$
