Cubic and quartic congruences modulo a prime

Let p>3 be a prime, and Np( f(x)) denote the number of solutions of the congruence f(x)≡0 (mod p). In this paper, using the third-order recurring sequences we determine the values of Np(x3+a1x2+a2x+a3) and Np(x4+ax2+bx+c), and construct the solutions of the corresponding congruences, where a1,a2,a3,a,b,c are integers.