This sequent is provable.

\begin{logicproof}{2}
    \begin{subproof}
        p \imp (q \land r) & \assum \\
        \begin{subproof}
            p & \assum \\
            q \land r & $\impe 1,2$\\
            q & $\ande{1} 3$
        \end{subproof}
        p \imp q & $\impi 2-4$ \\
        \begin{subproof}
            p & \assum \\
            q \land r & $\impe 1,6$ \\
            r & $\ande{2} 7$
        \end{subproof}
        p \imp r & $\impi 6-8$ \\
        (p \imp q) \land (p \imp r) & $\andi 5,9$
    \end{subproof}
    (p \imp (q \land r)) \imp ((p \imp q) \land (p \imp r)) & $\impi 1-10$
\end{logicproof}