Let $X_1, X_2, ..., X_n$ be a random sample from $N(\theta+2, \theta^2)$. Find the MLE of $\theta$.
I went through some work and I could not solve for $\theta$. Am I doing anything wrong?
$$L( \theta )= \prod _ { i = 1 } ^ { n } \frac { 1 } { \theta \sqrt { 2 \pi } } \exp \bigg[- \frac{( x _ { i } - \theta - 2 )^2}{2 \theta ^ { 2 }} \bigg] = \bigg( \frac {1 } { \theta\sqrt { 2 \pi } } \bigg )^n \exp \bigg [ - \frac { \sum _ { i = 1 } ^ { n } ( x _ { i } - \theta - 2 ) ^ { 2 } } { 2 \theta ^ { 2 } }\bigg ]$$
$$\ln [ L( \theta ) ] = - n \ln ( \theta \sqrt { 2 \pi } ) - \frac { \sum _ { i= 1 } ^ { n } ( x _ { i } - \theta - 2 ) ^ { 2 } } { 2 \theta ^ { 2 } }$$
$$\frac { d \ln [ L ( \theta ) ] } { d \theta } = - \frac { n } { \theta } - \frac{2 ( - 1 ) ( 2 \theta ^ { 2 } ) \sum _ { i = 1 } ^ { n } ( x _ { i } - \theta - 2 ) - 4 \theta \sum _ { i = 1 } ^ { n } ( x _ { i } - \theta - 2 ) ^ { 2 }}{4\theta ^4}$$
$$0 = \theta \sum _ { i = 1 } ^ { n } ( x _ { i } - \theta - 2 ) + \sum _ { i = 1 } ^ { n } ( x _ { i } - \theta - 2 ) ^ { 2 } - n \theta ^ { 2 }$$