Nice work and your conclusion is correct.
Further confirmation is the poles in the closed loop system are the roots of $$1+G(s)=0$$
Which results in the polynomial
$$s^2+(3+K)s+(2-K)$$
In order to have complex roots, using the quadratic formula $\sqrt{b^2-4ac}$ must be negative.
So we want to test if $4(2-K)>(3+K)^2$ for any K
Since $4(2-K)=(3+K)^2$ has roots:
$$K^2+4K+1=0$$
roots are: -3.732, -0.268
The range of K where the poles can be complex is therefore from -3.732 to -0.268