Let $M,N$ be modules over a Commutative ring $R$. If
$$0\to P_n \to P_{n-1}\to \cdots \to P_1 \to P_0 \to M\oplus N \to 0$$ is a finite projective resolution of $M \oplus N$, where $n\ge 1$, then how does one explicitly construct a finite projective resolution of $M$ without using Ext functor or without apriori knowing $M$ has finite projective dimension?