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To find the values of a, b, c, and d, we need to expand the given function F(x) and compare it with the general form of a cubic polynomial.

F(x) = (x-100)(2x-200)(4x-400) - (ax^3 + bx^2 + cx + d) F(x) = (x^2 - 200x + 10000)(4x - 400) - (ax^3 + bx^2 + cx + d) F(x) = 4x^3 - 400x^2 - 800x^2 + 80000x + 400x^2 - 40000x - ax^3 - bx^2 - cx - d F(x) = 4x^3 - 400x^2 - 800x^2 + 80000x + 400x^2 - 40000x - ax^3 - bx^2 - cx - d F(x) = 4x^3 - 1200x^2 + 40000x - ax^3 - bx^2 - cx - d F(x) = (4-a)x^3 + (-1200-b)x^2 + (40000-c)x - d

Comparing the coefficients of x^3, x^2, x, and the constant term with the general form of a cubic polynomial ax^3 + bx^2 + cx + d, we get:

4 - a = 0 -1200 - b = 0 40000 - c = 0 -d = 0

From the first three equations, we find that a = 4, b = -1200, and c = 40000. Substituting these values back into the original equation, we get:

F(x) = (x-100)(2x-200)(4x-400) - (4x^3 - 1200x^2 + 40000x) F(x) = (x^2 - 200x + 10000)(4x - 400) - 4x^3 + 1200x^2 - 40000x F(x) = 4x^3 - 400x^2 - 800x^2 + 80000x + 400x^2 - 40000x - 4x^3 + 1200x^2 - 40000x F(x) = 0

Therefore, the values of a, b, c, and d are a = 4, b = -1200, c = 40000, and d = 0.