Let \(P_{k+1}\) denote a path of length \(k\), let \(S_{m}\) denote a star with \(m\) edges, and let \(K_{n}(\lambda)\) denote the complete multigraph on \(n\) vertices in which every edge is taken \(\lambda\) times. In this paper, we prove that the necessary conditions are also sufficient for a \(\{P_{4}, S_{4}\}\)-decomposition of \(K_{n}(\lambda)\).