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The general idea is this theorem of High Algebra (of the theory of symmetric functions):
Every symmetric function of the roots of a polynomial is (= can be presented as) the function of the coefficients of the polynomial.
The rest is just technique. // Concretely, for your symmetric function, implementation of this idea might be technically
complicated and time consuming, and what is even worst - you will learn nothing useful from it.
By the way, from your post, the symmetric function is simply unreadable, so it does not leave me any possibility to implement this technique
even would I want to do it - but I don't want.
You can find some relevant examples in my lessons
- HOW TO evaluate functions of roots of a square equation
- HOW TO evaluate functions of roots of a cubic and quartic equation
in this site.
From these lessons, you will be able to see and to learn how this idea really works.