$\newcommand{\cov}{\mathrm{cov}}$ I'd like to ask a question similar to this. The question I've linked to is about $X = (X_1,X_2,\dots,X_n)^T$, a random vector in $\Bbb R^n$ and $A$, an $m \times n$ matrix. sebastianross asks how to find the covariance matrix of $AX$.
I'd like to ask about the covariance matrix of $2(\mathbf{b} + X^TB)$ where $\mathbf{b} \in \Bbb R^m$ and $B$ is an $n\times m$ matrix.
Imitating the answer to the question I've linked to,
$$\begin{aligned} \cov(X^TB) &= \mathbb{E} \left( (X^TB)(X^TB)^T \right) - \mathbb{E}(X^TB)\mathbb{E}(X^TB)^T \\ &= \mathbb{E}((X^TB)B^TX) - \mathbb{E}(X^T)B\mathbb{E}(B^TX) \\ &= \mathbb{E}((X^TB)B^TX) - \mathbb{E}(X^T)BB^T\mathbb{E}(X) \end{aligned} $$
(I'm leaving out the multiplication by $2$ and $\mathbf{b}$, although I'd like to go further at some point.)
So here, the common factor, $BB^T$, lies in the middle of the matrix products. What would you do next?