Answer
See below
Work Step by Step
We obtain:
$(T_2T_1)v=T_2(T_1(v))\\
=T_2(T_1(\alpha v_1 + \beta v_2))\\
=T_2(\alpha T_1( v_1)+\beta T_1(v_2))\\
= T_2(\alpha (v_1-v_2)+\beta (2v_1+v_2))\\
=T_2(\alpha v_1-\alpha v_2+2\beta v_1+\beta v_2\\
=T_2((\alpha+2\beta)v_1)+(\beta - \alpha)v_2)\\
=T_2(\alpha +2\beta)v_1 +T_2(\beta -\alpha)v_2\\
=(\alpha + 2\beta)(v_1+2v_2)+(\beta -\alpha)(3v_1-v_2)\\
=\alpha v_1+2\beta v_1+2\alpha v_2+4\beta v_2+3\beta v_1 -3 \alpha v_1-\beta v_2+\alpha v_2\\
=(\alpha +2\beta -3\alpha +3\beta)v_1+(2\alpha +4\beta +\alpha -\beta )v_2\\
=(-2\alpha +5\beta)v_1+(3\alpha +3\beta)v_2$
for $v=\alpha v_1 +\beta v_2$
