Provide a step-by-step mathematical proof demonstrating that the specific quintic equation x^5 - 4x + 2 = 0 is not solvable by radicals. Your proof must utilize Galois Theory. Start by calculating the discriminant to determine the nature of the roots, determine the Galois group of the polynomial over the rational numbers, and show that this group is not solvable. Explain the connection between the group's structure and the existence of a radical formula for the roots.