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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