Vieta's + Newton's = Trouble

Algebra Level 5

Consider the complex, non-zero, solutions a,b,ca, b, c and dd to

5x5+4x4+3x3+2x2+x=05x^5+4x^4+3x^3+2x^2+x=0

If the expression

ab+ac+ad+bc+bd+cd+abc+abd+acd+bcd+abcda2+b2+c2+d2abcd\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 mn-\frac{m}{n}, where mm and nn are positive, coprime, integers, find m+nm+n.

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