Fourth moment of a shifted normal

MediumProbability Random Variables Statistics~5m

Let $X \sim N(2, 1)$ - normal with mean $2$ and variance $1$ . What is the raw fourth moment $E[X^4]$ ?

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Write $X = 2 + Z$ with $Z\sim N(0,1)$ and expand $(2+Z)^4$ .
Odd moments of $Z$ vanish; use $E[Z^2]=1$ , $E[Z^4]=3$ .

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