The two north-south tracks are x=4 and x=-2 because these are the zeroes of x²-2x-8=(x-4)(x+2) in the denominator. The east-west track is y=2, the horizontal asymptote, which intersects the other two tracks at (4,2) and (-2,2).
One road is intersected by the track:
2=(2x²-2x-5)/(x²-2x-8),
2x²-4x-16=2x²-2x-5,
-4x-16=-2x-5,
-11=2x, so x=-11/2 and the intersection is (-11/2,2).