The characteristic equation of a closed loop system using a reverse acting proportional controller with gain Kc is:
12S^3 + 19S^2 + 8S + 1 + Kc = 0

What is the value of Kc which gives critically stable system?

 

The Kr that places the real part of the complex roots on the imaginary axis.

You can use Routh-Hurwitz or Root Locus methods.