Prove the identity in the title where $A$ is an $m\times n$ matrix of constants, $X$ is an $n\times 1$ random vector and $b$ is an $m\times 1$ constant vector.
I'm aware of the proofs for $E(AX+b) = AE(X)+b$ and $\text{Var}(AX)=A\text{Var}(X) A^T$, I just haven't seen a way to apply their logic here. Applying $Var(X)=\Bbb E(X^2)-[\Bbb E(X)]^2$ and expanding $[A\Bbb E(X)+b][A\Bbb E(X)+b]$ didn't lead me toward any other ideas either, since the dimensions of the matrix and vectors didn't allow further operations.
Update:
It seems that there was a printing error and this proof does not exist, as angryavian suggested.