f[m_,h_]:=(m^2-1)^2-m*h

Plot[{f[x,0],f[x,1.45],f[x,-1.45]},{x,-2,2}]

sol=Solve[D[f[m,h],m]==0,m]

InputForm[m/.sol[[1]]]

InputForm[m/. sol[[3]]]

InputForm[m/. sol[[2]]]

m1[h_]:=2/(3^(1/3)*(9*h+Sqrt[3]*Sqrt[-64+27*h^2])^(1/3))+(9*h+Sqrt[3]*Sqrt[-64+27*h^2])^(1/3)/(2*3^(2/3))
m2[h_]:=-((1-I*Sqrt[3])/(3^(1/3)*(9*h+Sqrt[3]*Sqrt[-64+27*h^2])^(1/3)))-((1+I*Sqrt[3])*(9*h+Sqrt[3]*Sqrt[-64+27*h^2])^(1/3))/(4*3^(2/3))
m3[h_]:=-((1+I*Sqrt[3])/(3^(1/3)*(9*h+Sqrt[3]*Sqrt[-64+27*h^2])^(1/3)))-((1-I*Sqrt[3])*(9*h+Sqrt[3]*Sqrt[-64+27*h^2])^(1/3))/(4*3^(2/3))

Plot[{m1[h],m2[h],m3[h]},{h,-2,2}]

Plot[{h+m1[h],h+m3[h]},{h,-2,2}]