This article develops an approach to achieving consensus while improving the dynamic performance for a class of homogeneous multi-agent systems (MASs) using delayed state information via eigenvalue assignment. Note that the distribution of roots of quasi-polynomials plays a fundamental role in the consensus protocol design of the MASs. Some necessary conditions for the distribution of roots for a class of quasi-polynomials are first derived. Then, these conditions are applied to estimate the allowable regions of the protocol parameters. Next, some necessary and sufficient conditions for the determination of effective protocol parameters are established. An illustrative example is provided to show the effectiveness of the designed protocols.