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In this paper, we address an open problem posed by Bai and Xia in [2]. We study polynomials of the form f (x) = x 4 q + 1 + λ 1 x 5 q + λ 2 x q + 4 over the finite field F 5 k , which are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on λ 1 , λ 2 ∈ F 5 k so that f(x) is a permutation monomial, binomial, or trinomial of F 5 2 k . [ABSTRACT FROM AUTHOR]