R-H range of K for K(s+13)/[s(s+3)(s+7)]

For unity feedback, the closed-loop characteristic equation is $1+G(s)=0$. The system is stable when the first column of the Routh array has no sign change.

Characteristic equation

$$\begin{aligned} 1 + \frac{K(s+13)}{s(s+3)(s+7)} &= 0 \\ s(s^2+10s+21) + Ks + 13K &= 0 \\ s^3 + 10s^2 + (21+K)s + 13K &= 0 \end{aligned}$$

Routh array

Conditions for stability

1. From the $s^1$ row: $\frac{210-3K}{10} > 0 \Rightarrow K < 70$
2. From the $s^0$ row: $13K > 0 \Rightarrow K > 0$

$$0 < K < 70$$

Marginal stability

At $K = 70$ the $s^1$ row vanishes. Auxiliary equation:

$$10s^2 + 13(70) = 0 \Rightarrow s^2 = -91 \Rightarrow s = \pm j9.54$$

So at $K=70$ the system oscillates at 9.54 rad/s, and for $K>70$ two poles move into the RHP.

Answer: the system is stable for $0 < K < 70$.