# Finding K values for all poles of real parts are less than -2

I have an open loop transfer function with k > 0 as below:

$$L(s) = \dfrac{k}{s(s^2+8s+25)}$$

Now for the closed loop transfer function, I would like to find the k value which makes all the real parts of poles less than -2. Below is how I get the closed-loop transfer function:

$$G(s) = \dfrac{L(s)}{1+L(s)} = \dfrac{k}{s^3+8s^2+25s+k}$$

Yet, I have no idea or method to locate the value k. I have used the Routh Criterion, and found that the boundary is $$200 > k > 0$$

However, this does not provide any useful information for me, is there any better way to solve this?

Let's make the variable change $$\u=s+2\$$ in the polynomial $$\P(s)=s^3+8s^2+25s+k\$$. The result is the polynomial $$Q(u)=P(u-2)=(u-2)^3+8(u-2)^2+25(u-2)+k=u^3 + 2u^2 + 5u + k - 26.$$ The Routh-Hurwitz criterion for this polynomial gives us the conditions $$\k-26>0\$$ and $$\10>k-26\$$, thus, $$\26.
Since $$\26 implies $$\{\rm Re}\; u<0\$$, we also have $$\{\rm Re}\; (s+2)<0\$$ and $$\{\rm Re}\; s<-2\$$.