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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