P3415
Denote $\mu : = \mathbb{E} X$. Using an add-and-substract trick with $- 2 \mu^T \mu + \mu^T \mu + \mu^T \mu = 0$, we have
\begin{equation}
\begin{split}
\mathbb{E} |X - a|^2
& = \mathbb{E}[(X - a)^T (X - a)] \\
& = \mathbb{E}(X^T X - X^T a - a^T X + a^T a) \\
& = \mathbb{E}(X^T X - 2 X^T a + a^T a) \\
& = \mathbb{E}(X^T X) - 2 \mathbb{E} (X^T a) + a^T a \\
& = \mathbb{E}(X^T X) - 2 \mu^T a + a^T a \\
& = \mathbb{E}(X^T X) - 2 \mu^T \mu + \mu^T \mu + \mu^T \mu - 2 \mu^T a + a^T a \\
& = \mathbb{E}[(X - \mu)^T (X - \mu)] + \mu^T \mu - 2 \mu^T a + a^T a \\
& = \mathbb{E}[(X - \mu)^T (X - \mu)] + (\mu - a)^T (\mu - a) \\
& = \mathbb{E} |X - \mu|^2 + |\mu - a|^2 \\
\end{split}
\end{equation}
Subtracting $|\mu - a|^2$ from both sides now yields the result. $\square$