R-H root distribution of s⁵+4s⁴+2s³+8s²+s+4

Result: RHP = 0, LHP = 1 (at $s=-4$), $j\omega$ axis = 4 ($\pm j1$, each twice). The system is unstable.

The number of sign changes in the first column gives the RHP roots. A zero row gives the roots that are symmetric about the origin; these are found from the auxiliary polynomial.

$$s^5+4s^4+2s^3+8s^2+s+4=0$$

Routh array

The $s^3$ row is all zeros. Auxiliary polynomial from the $s^4$ row:

$$A(s) = 4s^4+8s^2+4, \qquad \frac{dA}{ds} = 16s^3+16s$$

New auxiliary polynomial $A_1(s)=4s^2+4$, $\frac{dA_1}{ds}=8s$:

First column: $1,\ 4,\ 16,\ 4,\ 8,\ 4$. No sign change, so there is no root in the RHP.

Roots from the auxiliary polynomial

$$4s^4+8s^2+4 = 4(s^2+1)^2 = 0 \Rightarrow s=\pm j1,\ \pm j1$$

These 4 roots lie on the $j\omega$ axis (repeated pair). The remaining root: $\frac{s^5+4s^4+2s^3+8s^2+s+4}{(s^2+1)^2} = s+4$, so $s=-4$.

Stability

No root is in the RHP, but the roots on the $j\omega$ axis are repeated. They produce a response term like $t\sin t$, which grows with time. So the system is unstable.