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Algebra Level 5

For a fixed pair of real numbers \( (b, c) \) with \( b < 0 \), find the number of possible values of \(a\) that will satisfy the following conditions:

The equation \({ x }^{ 4 }+{ ax }^{ 3 }+{ bx }^{ 2 }+cx+d=0\) with \( a,b,c,d \in R\) has roots \({ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },{ x }_{ 4 }\) such that \({ x }_{ 1 }+{ x }_{ 2 }={ x }_{ 3 }+{ x }_{ 4 }\).

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