Series Demystified 2

Algebra Level 5

\[ \large \sum_{n=1}^{50} \left[ (n^3+3n^2+5n+2) \times (n+1)! \right] \]

If the summation above equals to \(S\), find the value of \( \frac{S+2}{52!} \).

This is one of my original Series Demystified problems.
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