Let the x-axis divide the segment joining $(-6,5)$ and $(-4,-1)$ in the ratio $k:1$ at point $(x,0)$. Using the section formula for the y-coordinate: $0 = \frac{k(-1)+1(5)}{k+1}$. $-k+5=0 \Rightarrow k=5$. So the ratio is $5:1$. Now find x-coordinate: $x = \frac{k(-4)+1(-6)}{k+1} = \frac{5(-4)+(-6)}{5+1} = \frac{-20-6}{6} = \frac{-26}{6} = -\frac{13}{3}$. Point of intersection $= \left(-\frac{13}{3}, 0\right)$.