Confused with region where root locus lies and sketch

Question

How will be the root locus of

$$G(s) = \frac K { (s^2+2s+2)(s^2+2s+5) } $$ look like?

The poles will be -1+2i, -1-2i, -1-i, -1+i which lies on the same vertical line and i am confused about the region where root locus lies in.

Answer 1

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 characteristic equation $1+G(s)$. The poles will move to be in different locations as $K$ is varied. The plot of this showing the trajectory of the poles versus $K$ is referred to as the root locus.