Quadratic Equations · Roots differing by a constant

Let

\alpha, \alpha + 2

\alpha \in \mathbb{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) = 4n

for some

n \in \mathbb{N}

. Then

(n + \alpha)

is equal to:

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