Complicated Quadratic Roots

Algebra Level 5

Let α and β be the roots of \(x^2 - 6x - 2 = 0\) with α>β and \(a_{n} = α^n - β^n\) and also let there be some quantity \(X = \frac{a_{10} - 2a_{8}}{2a_{9}}\). Also there is a quadratic equation \(x^2-(m-X)x+m=0\) where m is some real quantity.

Find number of all the possible integral values of m such that both the roots of this quadratic equation lie in the interval (1,2).

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