When sum converges

Algebra Level 4

\[\large \sum_{n=1}^{\infty} ~ \dfrac{1}{\sqrt[]{(n+2)(n^3+8+12n+6n^2)} - 4} = \dfrac{A}{B}\]

where \(A\) and \(B\) are coprime positive integers. Find the value of \(B-A\).

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