From Quadratic To Cubic

Algebra Level 3

Let \(a\) and \(b\) be the roots of the quadratic polynomial \(x^2-x-4=0.\) Let \(f(x)\) be a cubic polynomial such that \(f(a)=a, f(b)=b,\) \(f(a+b)=a+b,\) and \(f(ab)=-84.\) What is the value of \(f(6)\)?

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