For the quadratic x² + x + 1, the discriminant D = b² - 4ac = 1² - 4(1)(1) = -3. Since D < 0 and the leading coefficient (a=1) is positive, x² + x + 1 is always positive for all real x. Therefore, the sign of the whole expression depends solely on the denominator (x - 2). For the expression to be less than zero, x - 2 must be less than zero. Thus, x < 2, or x ∈ (-∞, 2).