F[c_]:=Module[{},M=200;a=-2;b=2;x0=0.7;eps=0.2;f[x_]:=x^4-x^2-c x;
r1=x/.NSolve[f'[x]==0,x]; r={};Do[If[r1[[k]]==Re[r1[[k]]],r=Append[r,r1[[k]]]],{k,Length[r1]}];roots=r;
A1={Black,Thickness[0.003],Line[{{-10-eps,0},{10+eps,0}}]};
A2={Black,Thickness[0.003],Line[{{0,-10-eps},{0,10+eps}}]};
A3={Red,Thickness[0.01],Line[Table[t=a+(b-a)k/M;{t,f[t]},{k,M}]]};
A8={Orange,PointSize[0.04],Table[Point[{roots[[k]],f[roots[[k]]]+0.03}],{k,Length[roots]}]};
A9={Blue,PointSize[0.07],Table[Point[{roots[[k]],f[roots[[k]]]+0.06}],{k,1}]};
S=Show[Graphics[{A9,A1,A2,A3}],AspectRatio->1,PlotRange->{{a,b},{-3.0,1.5}}];S];
Animate[If[c<1,F[c],F[2-c]],{c,0,1,0.01}]