= Solution
For an observed value $X=x$, the likelihood is
$$
p(x\mid\theta)=\frac{e^{-\theta}\theta^x}{x!},
$$
while the gamma prior with shape $\alpha$ and rate $\lambda$ has density
$$
\pi(\theta)=\frac{\lambda^\alpha}{\Gamma(\alpha)}
\theta^{\alpha-1}e^{-\lambda\theta},
\qquad \theta>0.
$$
The posterior density is therefore proportional to
$$
p(x\mid\theta)\pi(\theta)
\propto
\theta^{\alpha+x-1}e^{-(\lambda+1)\theta}.
$$
Normalizing this gamma kernel proves the <Poisson-gamma conjugacy> formula
$$
\boxed{\theta\mid X=x\sim
\operatorname{Gamma}(\alpha+x,\lambda+1)}.
$$
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