have a look at:
he mentions that total= a) homogenous+particular,
b) force + natural
c) zero state+ zero input
and that force+natural are not ...
I suspect that $G(s)$ is the Open Loop Gain. The root Locus would be the location of the poles in the Closed Loop Gain as you vary $K$. The poles in the Closed Loop Gain are given by the roots of the ...