Root loci having a total of four poles and zeros

Topological features are established for the root paths of algebraic equations of the form
N(s) + KM(s) = 0
, having a combined total of four poles (the
n
roots of
N(s) = 0)
and zeros (the
m
roots of
M(s) = 0)
. For
n \neq m
a simple unified treatment is given, based on intersections of a parabola with a rectangular hyperbola.