Center of mass of the system,

\({l_1} = \frac{{{m_1} \times 0 + {m_2} \times l}}{{{m_1} + {m_2}}} = \frac{{{m_2}l}}{{{m_1} + {m_2}}}\)

\({l_2} = l - {l_1} = \frac{{{m_1}l}}{{{m_1} + {m_2}}}\)

Required moment of inertia of the system,

\(I = {m_1}l_1^2 + {m_2}l_2^2\)
\(= \left( {{m_1}m_2^2 + {m_2}m_1^2} \right)\frac{{{l^2}}}{{{{\left( {{m_1} + {m_2}} \right)}^2}}}\)
\(= \frac{{{m_1} + {m_2}\left( {{m_1} + {m_2}} \right){l^2}}}{{{{\left( {{m_1} + {m_2}} \right)}^2}}} = \frac{{{m_1}{m_2}}}{{{m_1} + {m_2}}}{l^2}\)