Roots of a Quartic

Algebra Level 4

Let r1,r2,r3,r4r_1,r_2,r_3,r_4 be roots of the polynomial x44x3+6x2+8x+10x^4-4x^3+6x^2+8x+10.

1r2+r3+r4+1r1+r3+r4+1r1+r2+r4+1r1+r2+r3=AB\dfrac{1}{r_2+r_3+r_4}+\dfrac{1}{r_1+r_3+r_4}+\dfrac{1}{r_1+r_2+r_4}+\dfrac{1}{r_1+r_2+r_3}=\dfrac{A}{B}

where AA and BB are co-prime positive integers. Find A+BA+B.

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