Fun with triangles and polynomials

Geometry Level 5

It is given that \(a, b, c\) are sides of a triangle \(\Delta ABC\) and the roots of the equation below.

\[x^3-24x^2+180x-420=0.\]

If the value of \(\sin A+\sin B+\sin C=\dfrac{p}{q}\), where \(p, q\) are coprime positive integers, find \(p+q-10\).


Inspiration

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