= Solution
Assume finite second moments and write $Z_i=X_i-\mathbb EX_i$. Expanding the squared centered sum gives
$$
\begin{aligned}
\operatorname{Var}(X_1+X_2)
&=\mathbb E[(Z_1+Z_2)^2]\\
&=\mathbb E[Z_1^2]+\mathbb E[Z_2^2]+2\mathbb E[Z_1Z_2]\\
&=\operatorname{Var}(X_1)+\operatorname{Var}(X_2)+2\operatorname{Cov}(X_1,X_2).
\end{aligned}
$$
This <variance of a sum> identity does not require independence; the <covariance> term records the contribution that would otherwise be lost.
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