Binomial Number of Trade Batches

Let $N\sim\mathrm{Binomial}(5,0.4)$. Conditional on $N$, let \[ S=\sum_{i=1}^N B_i, \] where each batch size $B_i$ has PGF $H(s)=0.5+0.3s+0.2s^2$. Find the PGF of $S$ and compute $E[S]$.