Decompose mean squared error into bias and variance

Let $\hat{\theta}$ be an estimator of a parameter $\theta$, with mean squared error defined as $\operatorname{MSE}(\hat{\theta}) = \mathbb{E}[(\hat{\theta} - \theta)^2]$.

Show that $\operatorname{MSE} = \operatorname{Var}(\hat{\theta}) + \operatorname{Bias}(\hat{\theta})^2$, and give an example where a biased estimator beats an unbiased one on MSE.