The question quickly boils down to proving that each factor of the form $x^2+nx+m$, which has complex roots, can be multiplied by another factor which makes all the terms non-negative. Now, this can be done with an appropriate choice of the multiplied term. Just choose $g(x)=ax^2+bx+c$,where $a>0,c>0,b=lim_{n\to\infty}n$