Let p be a prime and let Fi belong to Zp[x1, x2,... , xp] be a symmetric polynomial, where Zp is the field of p elements. A sequence T in Zp of length p is called a Fi-zero sequence if Fi(T) = 0
a sequence in Zp is called a Fi-zero free sequence if it does not contain any Fi-zero subsequence. Define g(F, Zp) to be the smallest integer l such that every sequence in Zp of length l contains a Fi-zero subsequence. In this paper the author determine the value of g(Fi, Zp) and describe the set M(Fi, Zp) of all Fi-zero free sequences of maximal length g(Fi, Zp) - 1 for quadratic symmetric polynomials Fi of the form (x1 + x2 + ... + xp)2 + b(x2/1+x2/2+...+x2/p), where b belong to Zp, b diffrence 0, for the case p = 5.