$$\displaystyle{-2)-2)-2)-2}$$

Algebra Level 5

The coefficient of $$\displaystyle{x^2}$$ appearing after parenthesis have been removed and like terms have been collected in $\underbrace{\Big( ..............(x-2)^{2} -2)^{2} - 2)^{2}..)......\Big)^{2} }_\text{k times}$ can be expressed as $\dfrac{\alpha^{2k-1} - \alpha^{k-1}}{\beta}$. Where $$\displaystyle{\alpha}$$ and $$\displaystyle{\beta}$$ are positive coprime integers. Give their sum.

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