Its time for cubes

Algebra Level 5

Let \(a_1, a_2, \ldots,a_{4} \) denote the roots to the equation \( x^1 + 2x^2 + 3x^3 + 4x^4 = 0 \). Then,

\[ (8a_{ 1 }^{ 3 }+1)(8a_{ 2 }^{ 3 }+1)\cdots (8a_{ 4 }^{ 3 }+1)=-\frac { A }{ B } \]

where \(A,B\) are positive coprime integers. What is \(A+B\)?


Inspiration

You can refer to this

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