Show[RootLocusPlot[(TransferFunctionModelFull[((k (6+3 s+(s^2)))/(s(s+2) (10+5 s+(s^2)))), , , ]),{k,0,10000},PlotRange->{{-5,1},{-5,5}},PlotStyle->Directive[Thickness[0.014]] ,Frame->True,FrameStyle->Directive[Gray,Thickness[Small]]], FrameLabel->{{HoldForm["Im(s)"],None},{HoldForm["Re(s)"],None}},PlotLabel->None,LabelStyle->{22,GrayLevel[0]},AxesStyle->Gray]

Show[RootLocusPlot[(TransferFunctionModelFull[((k (10+2 s+(s^2)))/((s+0.5)(s+2)(s+3.5) (12.5+5 s+(s^2)))), , , ]),{k,0,100000},PlotRange->{{-30,15},{-30,30}},PlotStyle->Directive[Thickness[0.02]],Frame->True,FrameStyle->Directive[Gray,Thickness[Small]]], FrameLabel->{{HoldForm["Im(s)"],None},{HoldForm["Re(s)"],None}},PlotLabel->None,LabelStyle->{32,GrayLevel[0]},AxesStyle->Gray]

Show[RootLocusPlot[(TransferFunctionModelFull[((k (s+1)(s+3.5))/((s^2)(s+2)(s+2.5)(3+3 s+(s^2)))), , , ]),{k,0,100000},PlotRange->{{-4,2},{-5,5}},PlotStyle->Directive[Thickness[0.014]] ,Frame->True,FrameStyle->Directive[Gray,Thickness[Small]]], FrameLabel->{{HoldForm["Im(s)"],None},{HoldForm["Re(s)"],None}},PlotLabel->None,LabelStyle->{22,GrayLevel[0]},AxesStyle->Gray]

Show[RootLocusPlot[(TransferFunctionModelFull[((k (s+1))/(s((s^3)+3(s^2)+12s-16))), , , ]),{k,0,100000},PlotRange->{{-4,2},{-5,5}},PlotStyle->Directive[Thickness[0.014]] ,Frame->True,FrameStyle->Directive[Gray,Thickness[Small]]], FrameLabel->{{HoldForm["Im(s)"],None},{HoldForm["Re(s)"],None}},PlotLabel->None,LabelStyle->{22,GrayLevel[0]},AxesStyle->Gray]

D[(((s+1))/(s((s^3)+3(s^2)+12s-16))), s]==0//NSolve

kc = Solve[-832+59k-(k^2)==0, k]//N

(Sqrt[((3k)/(52-k))])/.kc

Show[RootLocusPlot[(TransferFunctionModelFull[((k (s+1)(s+2))/(s((s^3)+3(s^2)+12s-16))), , , ]),{k,0,100000},PlotRange->{{-4,2},{-5,5}},PlotStyle->Directive[Thickness[0.014]] ,Frame->True,FrameStyle->Directive[Gray,Thickness[Small]]], FrameLabel->{{HoldForm["Im(s)"],None},{HoldForm["Re(s)"],None}},PlotLabel->None,LabelStyle->{22,GrayLevel[0]},AxesStyle->Gray]

Show[RootLocusPlot[(TransferFunctionModelFull[((k (s+1))/(s(s+2)((s^3)+3(s^2)+12s-16))), , , ]),{k,0,100000},PlotRange->{{-4,2},{-5,5}},PlotStyle->Directive[Thickness[0.014]] ,Frame->True,FrameStyle->Directive[Gray,Thickness[Small]]], FrameLabel->{{HoldForm["Im(s)"],None},{HoldForm["Re(s)"],None}},PlotLabel->None,LabelStyle->{22,GrayLevel[0]},AxesStyle->Gray]