= Solution
Set $D(s)=4-2s-s^2$, so $D(s)G(s)=s^2$. Differentiate once and evaluate at $s=1$, using $D(1)=1$, $D'(1)=-4$, $G(1)=1$:
$$
-4+G'(1)=2,\qquad G'(1)=6.
$$
A second differentiation gives $D''G+2D'G'+DG''=2$, hence
$$
-2-8\cdot6+G''(1)=2,\qquad G''(1)=52.
$$
The derivatives of a <probability generating function> give $G'(1)=\mathbb E[F]$ and $G''(1)=\mathbb E[F(F-1)]$. Consequently \b[$\operatorname{Var}(F)=52+6-6^2=22$]. The earlier <geometric tail bound from a uniform escape probability> ensures the needed <moments> are finite.
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