Polynomials of degree 4 over finite fields representing quadratic residues

It is proved that in a finite field F of prime order p, where p is not one of finitely many exceptions, for every polynomial f(x) ∈ F [x] of degree 4 that has a nonzero constant term and is not of the form αg(x) there exists a primitive root β ∈ F such that f(β) is a quadratic residue in F . This refines a result of Madden and Vélez from 1982 about polynomials that represent quadratic residues at primitive roots.