You are given a transfer function $\displaystyle G(s)=\frac{1.81K(s+20)}{(s^3+10s^2+32s+32)}$. This system is connnected with unity negative feedback.
I've tried so many things but I can't do it . I've did $1.81K(s+20)=0$, but it's clearly wrong. I've got the zeros for the bottom part $($ $s=-2$ or $s=-4$ or $s=-4$).
So, $\frac{1.81K(s+20)}{s+2}\times \frac{1}{(s+4)^2}$.
Tried to equal the first one to zero, got nothing. I really don't know what to do .
The transfer function of the system with unit negative feedback is $\frac{G(s)}{1+G(s)}$. The denominator is $q(s) =s^3+10 s^2 + 32 s + 32 + 1.81 K (s + 20)$.
The solution is to look for the smallest value of $K$ that makes $q(s)$ have a root in the right half plane (i.e. the real part of a root is positive). You can do this numerically (try different values of $K$, say in increments of $10^-2$ from $0$ onwards), or use a root locus chart or something.
As pointed out in a comment, the Routh-Hurwitz stability criterion is another good option.