In this paper, we consider the quartic moment problem suggested by Curto-Fialkow[6]. Given complex numbers $gamma{equiv}{gamma}^{(4)};:;{gamma00},;{gamma01},;{gamma10},;{gamma01},;{gamma11},;{gamma20},;{gamma03},;{gamma12},;{gamma21},;{gamma30},;{gamma04},;{gamma13},;{gamma22},;{gamma31},;{gamma40}$, with ${gamma00},;>0;and;{gamma}_{ji}={gamma}_{ij}$ we discuss the conditions for the existence of a positive Borel measure ${mu}$, supported in the complex plane C such that ${gamma}_{ij}=int;={z}^i;z^j;d{mu}(0{leq}i+j{leq}4)$. We obtain sufficient conditions for flat extension of the quartic moment matrix M(2). Moreover, we examine the existence of flat extensions for nonsingular positive quartic moment matrices M(2).