= Solution
Let $s=\sum_ix_i$, $A=a+s$ and $B=b+n$. Conditional independence gives the <Poisson distribution>'s <likelihood>
$$
L(\theta;x)=\frac{\theta^s e^{-n\theta}}{\prod_ix_i!}.
$$
Multiply by the prior <gamma distribution>'s density and discard factors independent of $\theta$:
$$
\pi(\theta\mid x)\propto\theta^{a+s-1}e^{-(b+n)\theta}.
$$
Normalizing with the <gamma function> identifies
$$
\boxed{\theta\mid x\sim\operatorname{Gamma}(A,B),\qquad
\pi(\theta\mid x)=\frac{B^A}{\Gamma(A)}\theta^{A-1}e^{-B\theta},\quad\theta>0.}
$$
Here $A=a+s$ is the shape and $B=b+n$ is the rate, not the scale. This is <conjugate prior> updating for a <Poisson distribution>.
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