= Solution
Writing $s=\sum_i x_i$, the <Poisson distribution> likelihood is proportional to $\theta^s e^{-n\theta}$. Multiplying by the <gamma distribution> prior gives
$$
\pi(\theta\mid\boldsymbol x)\propto\theta^{r+s-1}e^{-(\lambda+n)\theta}.
$$
This is <Poisson-gamma conjugacy>, with the rate convention made explicit:
$$
\boxed{\theta\mid\boldsymbol x\sim\operatorname{Gamma}(r+s,\text{rate }\lambda+n)}.
$$
Its normalized density is $(\lambda+n)^{r+s}\theta^{r+s-1}e^{-(\lambda+n)\theta}/\Gamma(r+s)$ for $\theta>0$.
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