JEE Advanced algebra problem

Algebra Level pending

Let there be a polynomial \(ax^4 + bx^3 + cx^2 + dx + e \) such that \(a+e=5, ae = 6, e>a\). And \(a,b,c,d\) and \(e\) are all real numbers. Let \( \alpha, \beta , \gamma\) and \(\delta\) be roots of the above polynomial. Find

\[ (b^2-ac) \dfrac{S_{10}}{S_{12}} + (bc-ad) \dfrac{S_9}{S_{12}} + (bd-ae) \dfrac{S_8}{S_{12}} + be \dfrac{ S_7}{S_{12}} , \]

where \(S_n = \alpha^n + \beta^n + \gamma^n + \delta^n\), where \(n\) is a positive integer.

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