Find the surface satisfying $\frac{\partial ^2 z}{\partial x^2} = 6x+2$ and touching $z=x^3+y^3$ along its section by the plane $x+y+1=0$

Question

I started off by staring to solve for the DE first:

as $$z=x^3+x^2+C(y)+D$$ where C is a function of y and D is an arbitrary constant of integration; and this is all I could do. Kindly provide any hints as to what it means 'touching (some surface) along its section (by a plane)', as I am not able to proceed after this.