A=3;m=1;β=2;ω1=8;

f[t_]=A t^2 HeavisideTheta[(2-t)(t+2)];

p1=Plot[f[t],{t,-3,3},PlotStyle->Red]

g[t_,tp_]=HeavisideTheta[t-tp] 1/(m ω1)Exp[-β (t-tp)] Sin[ω1(t-tp)];

x[t_]=Integrate[f[tp] g[t,tp],{tp,-Infinity,t}];

p2=Plot[50x[t],{t,-4,7},PlotStyle->Green,PlotRange->All]

Show[p1,p2,PlotRange->All]