Below is a problem in an degree exam in real analysis:
$$\int\_{\Bbb R^2}e^{-(4x^2+4xy+5y^2)}d(x,y)=?$$

The original idea by me is to diagonalize such quadratic form. The result of diagonalization of the quadratic form $4x^2+4xy+5y^2$ is: [![enter image description here](https://i.stack.imgur.com/6fINL.png)](https://i.stack.imgur.com/6fINL.png)

However, the eigenvalues are very ugly. How to do the rest step to compute the integral? Is there an easy way?