Vieta's Derivative II -- My 500-follower problem

Calculus Level 4

The function x5+6x418x310x2+45x24x^5 + 6x^4 - 18x^3 -10x^2 + 45x -24 has only four distinct real roots: α\alpha, β\beta, γ\gamma and δ\delta (in no particular order). If f(x)f'(x) is the first derivative of f(x)f(x). Evaluate

f(α)+f(β)+f(γ)+f(δ)f'(\alpha) + f'(\beta) + f'(\gamma) + f'(\delta)

Inspiration: This is exactly the problem titled Inspired by Vieta's Derivatives with a twist.

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