Answer by CroCo for Root Locus, Gain in Feedback Loop. Need help

In order to draw the root locus, you need to convert the open-loop system into the closed-loop system. You can do this

$$ \begin{align} T(s) &= \frac{ \frac{1}{(s+3)(s+4)} }{ 1+ \frac{1}{(s+3)(s+4)} \frac{K}{s(s+1)} } \\ &= \frac{s^2+s}{s^4+8s^3+19s^2+12s+K} \end{align} $$ Now the gain K appears in the characteristic equation of the closed-loop \$T(s)\$ (i.e the denominator). You can easily vary K from 0 to infinity to compute the poles of the characteristic equation of the closed-loop \$T(s)\$ using any software (e.g. Matlab). If the gain K is zero, we have these poles 0,-1,-3,-4. Use this Matlab script to compute the poles at different gains

K=0; % vary it from 0 to big number
poles = roots([1 8 19 12 K])

You just need tabular data to plot the root locus. You may use Matlab as well.

GH=zpk([],[0 -1 -3 -4],1);
sys=tf(GH);
rlocus(GH)

Or this elegant code

s=tf('s');
G = 1/((s+3)*(s+4));
H = 1/(s*(s+1));
sys = G*H;
rlocus(sys)

Let's go back to roots([1 8 19 12 K]) if K is 178, we have these poles

  -4.8276 + 2.3442i
  -4.8276 - 2.3442i
   0.8276 + 2.3442i
   0.8276 - 2.3442i

We can see same result in the root locus obtained by Matlab at this gain,