if one of the circles x^2+y^2+2g1x+c=0 and x^2+y^2+2g2x+c=0 lies within the other then prove that g1g2>0 and c>0
If $\displaystyle c\leqslant0$ then the point $\displaystyle (0,\sqrt{-c})$ lies on both circles. In that case, the circles intersect, so one of them cannot lie inside the other. Conclusion: c must be greater than 0.
So now assume that c>0. Check that in this case, neither circle contains any points on the y-axis. So each circle (and its interior) lies entirely on one side of the y-axis. Your next step is to find where the centres of the circles are. If they are on opposite sides of the y-axis then neither circle can lie inside the other. So the centres must lie on the same side of the y-axis. Deduce that $\displaystyle g_1$ and $\displaystyle g_2$ have the same sign and therefore $\displaystyle g_1g_2>0$.