Dare to find my roots! 2

Algebra Level 4

\[x^5+5\lambda x^4-x^3+(\lambda \alpha -4)x^2-(8\lambda+3)x+(\lambda \alpha -2)=0 \\ \lambda,\alpha \in \mathbb{R} \]

Find the value of \(\alpha\) for which the above equation has exactly two roots independent of \(\lambda\).

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