root locus of \$t(s)=\frac{K(s+2)}{(s+3)(s^2+2s+2)}\$?
A breakaway point can only occur between two adjacent real open-loop poles. A break-in point can only occur between two adjacent real open-loop zeroes.

A real solution to \$\frac{dK}{ds}=0\$ is not a sufficient condition to establish the existence of a breakaway or break-in point.