Rooting out the answer

Algebra Level pending

Let \(Ax^4 + Bx^3 + Cx^2 + Dx + E= 0 \) be an equation with integer coefficients such that \(A\) is positive, \(\gcd(A,B,C,D,E) = 1\), and the roots are \(\sqrt2,-\sqrt2,\sqrt3,-\sqrt3\).

Find \(A+B+C+D+E\).

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