what values of c is the locus of the function (x^2+x+y+3)y+z(x+y+3)+c a surface?

ok so my f=yx^2+yx+y^2+3y+zx+zy+3z+c..

i know that on a surface df cant equal zero.. and

{x,y,z in R^3 : f(x,y,z)=0}

df=(2xy+y+z)dx+(x^2+x+2y+z+3)dy+(x+y+3)dz

2xy+y+z=0

x^2+x+2y+z+3=0

x+y+3=0

im confused now.. am i meant to solve them? coz i get very bad answers