UMVUE and CRLB for normal parameter

Let $X_1, X_2, X_3, X_4$ be a random sample of size $4$ from an $N(\theta, 1)$ distribution, where $\theta\in\mathbb{R}$ is an unknown parameter. Let $\bar{X}=\frac{1}{4}\sum_{i=1}^4 X_i$, $g(\theta)=\theta^2+2\theta$ and $L(\theta)$ be the Cramer-Rao lower bound on variance of unbiased estimators of $g(\theta)$. Then, which one of the following statements is FALSE?