How to put a bivariate normal distribution under standard form

Question

If I have the following gaussian integral:

$$ \int_{-\infty}^{a}\int_{-\infty}^{b}\exp\left[-\alpha x^2+\beta x-\theta y^2+\gamma y+2\lambda xy\right] dxdy$$ ,where X and Y are standard normal R.V.

Well I can put it under the form:

$$ \exp\left(\frac{\beta^2}{4\alpha} +\frac{\lambda^2}{4\theta}\right)\int_{-\infty}^{a}\int_{-\infty}^{b}\exp\left[-(\sqrt\alpha x-\frac{\beta}{2\sqrt\alpha})^2 -(\sqrt\theta y-\frac{\gamma}{2\sqrt\theta})^2+2\lambda xy\right] dxdy$$

But from then... I don't know how to reach a bivariate normal distribution form... and my head starts to hurt.

More precisely, when doing the change of variable. In my second eqn, say I def $\hat{x}:=(\sqrt\alpha x-\frac{\beta}{2\sqrt\alpha})$ and $\hat{y}:=(\sqrt\theta y-\frac{\gamma}{2\sqrt\theta})$, then after shifting limits and adjusting the differentials, I'm left with: $$ C\int_{-\infty}^{a*}\int_{-\infty}^{b*}\exp\left[-\hat{x}^2-\hat{y}^2+2\lambda(\hat{x}+...)(\hat{y}+...)\right] d\hat{x}d\hat{y}$$

And now I have some additional x and y double factors that I don't know what to do with...

It looks like this actually http://en.wikipedia.org/wiki/Gaussian_integral#n-dimensional_with_linear_term (n-dimensional with linear term), and I have no problem finding matrix $A$ and vector $B$ that fit my expression but they just go straight to the solution for integration over $R^2$ and give no link to a bivariate...

I still take any hint. Thanks