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This sequent is provable.
\begin{logicproof}{1}
(p \land q) \lor (p \land r) & \prem \\
\begin{subproof}
p \land q & \assum \\
p & $\ande{1} 2$
\end{subproof}
\begin{subproof}
p \land r & \assum \\
p & $\ande{1} 4$
\end{subproof}
p & $\ore 1, 2-3, 4-5$ \\
\begin{subproof}
p \land q & \assum \\
q & $\ande{2} 7$ \\
q \lor r & $\ori 8$
\end{subproof}
\begin{subproof}
p \land r & \assum \\
r & $\ande{2} 10$ \\
q \lor r & $\ori 11$
\end{subproof}
q \lor r & $\ore 1, 7-9, 10-12$ \\
p \land (q \lor r) & $\andi 6, 13$
\end{logicproof}