Ly = LaplaceTransform[y''[t] + 3y'[t] + 2y[t],t,s]

Lrhs = LaplaceTransform[UnitStep[t-2]Exp[t-2],t,s]

eq = Ly==Lrhs

simplifiedeq = eq /. {y[0]-> 0,y'[0]->1,LaplaceTransform[y[t],t,s]->Y[s]}

solinsdomain = Solve[simplifiedeq,Y[s]]

simplifiedeq

y[t] = InverseLaplaceTransform[Y[s] /. solinsdomain, s, t]