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\begin{align*} (n+1)^3 + 2(n+1) &= n^3 + 3n^2 + 3n + 1 + (2n +2)\\ &= n^3 + 2n + 3n^2 + 3n + 3 \\ &= (n^3 + 2n) + 3(n^2 + n + 1) \end{align*}