Using the quadratic formula, the trinomial x² - 2x - 1 does not factor into integer values and is irreducible over the integers, hence does not match the provided options.
The student is requested to use the quadratic formula to factor the trinomial x² - 2x - 1. To use this method, the equation should be in the form ax² + bx + c = 0. The quadratic formula is √(b² - 4ac)/2a. In this case, a = 1, b = -2, and c = -1.
Plugging these values into the quadratic formula yields the solutions for x. However, factoring the trinomial directly would require finding two numbers that multiply to -1 (a*c) and add to -2 (b). When using the quadratic formula, we find that the solutions do not result in factors with integer values, implying that the trinomial is irreducible over the integers and does not factor as any of the options provided.