$$G_X(t)=e^{(t+2)(t-1)}$$
Is it true that $X$ follows $Po(t+2)$ then? Because the generating function can be rewritten as
$$G_X(t)=\sum_{x=0}^∞ t^x\frac{e^{-t-2}(t+2)^x}{x!}$$
and then there would be an inconsistency with $$E(X)=G_X'(1)=3≠t+2$$
There would also be an inconsistency with
$$Var(X)=G_X''(1)+G_X'(1)-[G_X'(1)]^2=5≠t+2$$
and
$$P(X=1)=G_X'(0)=e^{-2}≠e^{-t-2}(t+2)$$
Someone help me with my misconception?