Irreducible Quadratic Factors ofx(qn+1)/2+ax+bover q

Letf(x) =x(qn+1)/2+ax+b q[x] withb≠ 0,q=ptandpan odd prime. Ifnis even ora2+ 1 is a square in qthenf(x) does not have an irreducible quadratic factor in q[x]. Iff(x) has a monic irreducible quadratic factor in q[x] then it is unique and equal tox2+ 2(b/a)x+b2/(a2+ 1). A condition thatx2+ 2(b/a)x+b2/(a2+ 1) dividef(x) is expressed in terms of quadratic and biquadratic symbols which are evaluated fora= ±1, 0 and allq, ora= ±2, ±3 andq=p.