Let $f:M_\bullet \to N_\bullet$ be a quasi-isomorphism between chain complexes of flat $R$-modules for $R$ a unitary ring. Let $P$ be a right $R$-module. Is it true, and how could I prove that $$f\otimes_R P:M_\bullet \otimes_R P \to N_\bullet \otimes_R P$$ is a quasi-isomorphism?
If necessary, one may assume that $M_\bullet$ and $N_\bullet$ are bounded.