The problem consists of two questions: a) Find k so that one of the roots is at s=-2 $$(s+1)^5+k=0$$ b) for the value of k you just found find the rest roots using root locus.
Regarding the first question s=-2 can be a solution if k=1. Whenever I use root locus there is a parameter in the equation. Since we consider k=1 in the b question how am I supposed to draw a root locus?
Update: This is a problem from an exam sheet and computers can't be used. What are the ways of finding the 4 other roots by hand? I can draw the root locus easily and I can see the s=-2 pole's movement and position when k becomes 1. What happens in the other 4 branches(poles) of the root locus though? Has the pole travelled the same distance in each of them?
I could also see that we have 2 conjugate pairs of poles but the system is too complicated to solve.