Assume that $\bf{x}$ is a random vector that is distributed multivariate normal with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. Let $\bf{A}$ be a matrix of constants.
I'm trying to prove that
$$ \operatorname{Cov}[\bf{x}, \bf{x}^T \bf{A} \bf{x}] = 2 \boldsymbol{\Sigma} \bf{A} \boldsymbol{\mu} $$
In the process of simplifying the expression, I've encountered the expectation of the product:
$$ \operatorname{E}[\bf{x} \bf{x}^T \bf{A} \bf{x}] $$
Given the assumption of a MVN distribution, can this be simplified?