To find where the lines do not intersect, we find first where they do intersect; the solution will be the other values of c.
Applying the Quadratic Formula, we get:
So there will be valid solutions (and thus intersections) for (c + 2)(c - 10) > 0. To find the solution, you need to solve the quadratic inequality.
The zeroes of (c + 2)(c - 10) are obviously at c = -2 and c = 10. The parabola corresponding to y = (c + 2)(c - 10) will open upward, so the quadratic will be positive (that is, the parabola will be above the x-axis) on the ends, before the first zero and after the second zero.
So which intervals give intersections? Then which interval does not?