Given two multivariate gaussians distributions, given by mean and covariance, $G_1(x; \mu_1,\Sigma_1)$ and $G_2(x; \mu_2,\Sigma_2)$, what are the formulae to find the product i.e. $p_{G_1}(x) p_{G_2}(x)$ ?

And if one was looking to implement this in c++, what would an efficient way of doing it?

Go easy, I am primarily a computer scientist and not a pure mathematician.

Any help much appreciated.