The probability distribution of a random variable $x$ is given below: \[\] $\begin{array} {llllllll} \textbf{X:}& 0& 1& 2& 3 \\ \textbf{P(X=x):}& 0.1& 0.3 &0.5& 0.1& \end{array}$\[\] If $Y=X^{2}+2X$ find the mean and variance of $Y$.

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