How many real solutions does the system y = -x² + 4x + 1 and y = x - 2 have?

Set the two expressions for $y$ equal: $-x^2 + 4x + 1 = x - 2$. Move everything to one side: $0 = x^2 - 3x - 3$. The discriminant is $b^2 - 4ac = (-3)^2 - 4(1)(-3) = 9 + 12 = 21$. Since it is positive, the quadratic has two distinct real roots, so the parabola and the line meet at 2 points, choice B. Choice A (0) would need a negative discriminant. Choice C (1) would need a discriminant of zero, meaning tangency. Choice D (4) is impossible, since a line and a parabola intersect at most twice.