Prove all lines parallel to an asymptote of a hyperbola intersect the hyperbola once only

You don't have a quadratic equation, $c^2+4cx+16=0$ is a linear equation with the single solution $x=-\frac{16+c^2}{4c}$, so there is only one intersection point when $c \ne 0$ and none when $c = 0$.

You are actually done at $$c^2+4cx+16=0$$

Remember that you're solving for $x$, and you get only one such $x$, unless $c=0$.