+9519 Observation: What's the minimal polynomial of \(x\)?
\(2x = 1 + \sqrt 3\\ 2x - 1 = \sqrt 3\\ (2x - 1)^2 = 3\\ 4x^2 - 4x - 2 = 0\\ 2x^2 - 2x - 1 = 0\)
Having that in mind:
\(\quad 4x^3 + 2x^2 - 8x + 7\\ = 4x^3 - 4x^2 - 2x + 6x^2 - 6x + 7\\ =2x(2x^2 - 2x - 1) + 3(2x^2 - 2x - 1) + 10\\ = 2x(0) + 3(0) + 10\\ = \boxed{10}\)
