The necessary and sufficient condition for the quinary cyclic codes with generator polynomial (x + 1)m _α (x)m _α ^e (x) to have parameters [5 ^m - 1,5 ^m - 2m - 2,4] is provided by analyzing solutions of certain equations over the finite field 𝔽₅ ^m . And thus several classes of new optimal quinary cyclic codes with the same parameters and generator polynomial are constructed based on analyzing irreducible factors of certain polynomials with low degrees over finite fields.