Is there an obvious trick I am missing for solving the following integral:

$$ \int_x P(y|x) W(x) (-x^TMx+2x^Tm -c)dx$$

Distributions are Gaussians and $M$ is symmetric.

I know how to do the expectation for a univariate Gaussian but I'm not sure of this multivariate case since $$ P(y|x) W(x) = V(y,x) $$

$W$ is the PDF of $x$, $P$ is a conditional PDF of $y$ given $x$ and $V$ is the joint PDF of $x$ and $y$.