Sum of roots with a twist

Calculus Level 3

Let αn,βn\alpha_{n}, \beta_{n} be the distinct roots of the equation x2+(n+1)x+n2=0x^{2}+(n+1)x+n^{2} = 0. If

n=220151(αn+1)(βn+1)\sum_{n=2}^{2015} \frac{1}{(\alpha_{n}+1)(\beta_{n}+1)}

can be expressed in the form ab\frac{a}{b}, where aa and bb are positive coprime integers, find bab-a.

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