In this paper we extend the notion of quasi-multipliers to the dual of a Banach algebra $A$ whose second dual has a mixed identity. We consider algebras satisfying weaker condition than Arens regularity. Among others we prove that for an Arens regular Banach algebra which has a bounded approximate identity the space $QM_{r}(A^{*})$ of all bilinear and separately continuous right quasi-multipliers of $A^{*}$ is isometrically isomorphic to $A^{**}.$ We discuss the strict topology on $QM_{r}(A^{*})$ and apply our results to $C^{*}-$algebras and to the group algebra of a compact group.