To find the roots of the quadratic function f(x) = -x² + 4x + 2, we can apply the quadratic formula, which is x = (-b ± √(b² - 4ac)) / 2a. Here, the coefficients are a = -1, b = 4, and c = 2. First, we calculate the discriminant (b² - 4ac): 4² - 4(-1)(2) = 16 + 8 = 24. Next, we substitute these values into the quadratic formula: x = (-4 ± √24) / 2(-1). Simplifying this gives us x = (-4 ± 2√6) / -2, which further simplifies to x = 2 ± √6. This results in two roots: Root 1 is approximately -0.4494897427831779 and Root 2 is approximately 4.449489742783178.