Let \(\alpha ,\alpha +2,\alpha \in \mathrm{ℤ}\), be the roots of the quadratic equation \(x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\…

Let \(\alpha ,\alpha +2,\alpha \in \mathrm{ℤ}\), be the roots of the quadratic equation \(x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\ldots\)\(+(x+n-1)(x+n+1)=4 n\) for some \(n \in \mathbb{N}\). Then \(n+\alpha\) is equal to:

[JEE Main 2026, 2 Apr (Shift 1)]