Long but lovely equation

Algebra Level 5

\[\dfrac{(x+1)(x-3)}{5(x+2)(x-4)}+\dfrac{(x+3)(x-5)}{9(x+4)(x-6)}-\dfrac{2(x+5)(x-7)}{13(x+6)(x-8)}=\dfrac{92}{585}.\]

If the real root of the above equation is of the form \(a\pm\sqrt{b}\),find \(a+b\).

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