Ugly Roots, Nice Answer

Algebra Level 3

Let the roots of the polynomial \[ 2x^{3}-3x^{2}+5x+c\] (where \(c\) is a constant) be \(\alpha\), \(\beta\), and \(\gamma\). If \[\frac{1}{\alpha\beta}+\frac{1}{\alpha\gamma}+\frac{1}{\beta\gamma}=\alpha\beta+\alpha\gamma+\beta\gamma,\] then the value of \(c\) can be written as \(-\frac{m}{n}\), where \(m\) and \(n\) are positive, coprime integers. Find the value of \(m+n\).

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