Vieta's + Newton's = Trouble

Algebra Level 5

Consider the complex, non-zero, solutions \(a, b, c\) and \(d\) to

\[5x^5+4x^4+3x^3+2x^2+x=0\]

If the expression

\[\dfrac{ab+ac+ad+bc+bd+cd+abc+abd+acd+bcd+abcd}{a^2+b^2+c^2+d^2-abcd}\]

can be written as \(-\frac{m}{n}\), where \(m\) and \(n\) are positive, coprime, integers, find \(m+n\).

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