How do I get started?Find the points on the surface at which the tangent plane is parallel to the plane .
Parallel planes have the same normal vector (i.e. $\displaystyle \langle 3,3,-1\rangle$). The constant term is irrelevant.
The equation for the tangent plane is
$\displaystyle F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0$
$\displaystyle F_x=2x$
$\displaystyle F_y=6y$
$\displaystyle F_z=8z$
So you want $\displaystyle 2x_0=3k$, $\displaystyle 6y_0=3k$, and $\displaystyle 8z_0=-k$ (because you have to take into account the normal vector can be scaled by any constant $\displaystyle k$).
Spoiler: