= Solution
For any action $a$, decompose the posterior expected <quadratic loss> around the <posterior mean>:
$$
\mathbb E[(a-\theta)^2\mid X]
=(a-\mathbb E[\theta\mid X])^2+\operatorname{Var}(\theta\mid X).
$$
The unique minimizer is therefore the <Bayes estimator under squared error loss>. Using the mean of the posterior <Beta distribution> gives
$$
\boxed{\widehat\theta=\mathbb E[\theta\mid X]
=\frac{S+1}{n+2}}.
$$
Under repeated sampling at parameter value $\theta$, $S$ has the <binomial distribution> with expected value $n\theta$. Consequently
$$
\boxed{\mathbb E_\theta[\widehat\theta]=\frac{n\theta+1}{n+2}},
$$
so its <bias of an estimator> is $(1-2\theta)/(n+2)$.
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