Does gain affect stability of a Transfer Function?
This system has 6 poles and one zero.

fully factored it becomes:

\$ \dfrac{K(2s+3)}{(s-(0.329881+1.27963 i)) (s-(0.329881-1.27963 i)) (s+(1.32988-1.49665 i)) (s+(1.32988+1.49665 i))}\$

Zeros:

\$s=\dfrac{-3}{2}\$

Poles:

\$2^{nd}\$ order at \$s=0\$

\$1^{st} \$ order @ \$s=  0.329881+1.27963 i\$

\$1^{st} \$ order @ \$s=  0.329881-1.27963 i\$ 

\$1^{st} \$ order @ \$s=  -1.32988+1.49665 i\$ 

\$1^{st} \$ order @ \$s=  -1.32988-1.49665 i\$ 

So this is non-causal, unstable in a strict sense.  However, someone who has more recent practise in this area should be able to use the info above and tell you the limits.  I would be answering "It isn't stable anyways so the K factor doesn't matter" but then I'm rusty but I did the factoring so I thought I'd put it up here for others to use.