I have a third-order polynomial (i.e., characteristic equation) for a proportional controller. In attempting to find the ultimate (or critical) gain, $K_c$, this is compared with a generic equation. My polynomial, I guess, is arbitrary but here is it anyway: $$ s^3+4s^2+4s+4K_{C} $$ The coefficients are compared directly to the expansion of this equation: $$(s^2+\omega^2)(s+a)$$
My questions are specifically: where does this generic form [i.e., $(s^2+\omega^2)(s+a)$] come from and why does this work?