Since $\mathbb{E} |X|^2 < \infty$, result R3581: Absolute moment inherits finiteness from greater exponents for random complex number guarantees that also $\mathbb{E} |X| < \infty$. Additionally, result R4514: Even integer absolute moments coincide with moments for random basic real number shows that $\mathbb{E} |X|^2 = \mathbb{E} X^2$, whence $\mathbb{E} X^2 < \infty$ as well. That is, all the relevant quantities exist and are finite.

Then, by direct computation, we have \begin{equation} \begin{split} \mathsf{Var}(X) & = \mathbb{E}[(X - \mathbb{E} X)^2] \\ & = \mathbb{E}(X^2) - \mathbb{E}(2 X \mathbb{E} X) + (\mathbb{E} X)^2 \\ & = \mathbb{E}(X^2) - 2 \mathbb{E} (X) \mathbb{E}(X) + (\mathbb{E} X)^2 \\ & = \mathbb{E}(X^2) - 2 \mathbb{E} (X)^2 + (\mathbb{E} X)^2 \\ & = \mathbb{E}(X^2) - (\mathbb{E} X)^2 \end{split} \end{equation} $\square$