S = {{-1.5,1.26},{-0.5,1.59},{-0.25,1.24},{0.25,0.72},{0.75,0.21},{1.00,0.14},{1.5,0.77},{2.00,4.86}}

A =Table[{1, Exp[-S[[i,1]]],Exp[S[[i,1]]],Exp[-2*S[[i,1]]],Exp[2*S[[i,1]]]},{i,1,8}]

A//MatrixForm

(B = Drop[S,{},{1}])//MatrixForm

Inverse[Transpose[A].A].Transpose[A].B//MatrixForm

F  = 1.12797+0.90324812331199*Exp[-x]-1.0546*Exp[x]-0.183721*Exp[-2x]+0.208836*Exp[2x]

FitPlot = Plot[F,{x,-1.75,2.25}, PlotStyle->Black];

PtPlot = ListPlot[S];

Show[FitPlot,PtPlot]