Let α, α+2, α ∈ Z, be the roots of the quadratic equation x(x+2)+(x+1)(x+3)+(x+2)(x+4)+….+(x+ n -1)(x+ n +1)=4 n for some n ∈ N. Then n +α is equal…

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