Dare to find my roots!

Algebra Level 5

\[x^5+5\lambda x^4-x^3+(\lambda \alpha -4)x^2-(8\lambda+3)x+(\lambda \alpha -2)=0 \\ \lambda,\alpha \in \mathbb{R} \] If the value of \(\alpha\) for which the above equation has exactly one root independent of \(\lambda\) is in the form \(-\dfrac{a}{b}\) where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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