Sum of roots with a twist

Calculus Level 3

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

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

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