Algebra or Number Theory?

\[ \big(n^2 + 2n + 2\big) \, \Big| \, \big(n^3 + 4n^2 + 4n - 14\big)\] Find the sum of all integers \(n\) such that the above holds true.

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Notation: \(a \mid b\) means \(a\) is a factor of \(b\).

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