A wonky quartic

Algebra Level 5

The sum of all values of aa such that the equation (x2x+a+1)2=4a(5x2x+1)(x^2-x+a+1)^2=4a(5x^2-x+1) has exactly three distinct real solutions, is of the form n+km\dfrac{n+\sqrt{k}}{m}, where k,m,nk,m,n are integers, k0,k\geq 0, m1m\geq 1 and mm is the smallest possible. Find k+n+m.k+n+m.

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