All you do is use the quadratic formula and look at the discriminant. So,
b = (-m-n) c = -p^2, to show that the roots are always real, we must show that the quantity (-m-n)^2 +4p^2 is always nonnegative for any real numbers. Obviously, the sum of two squares of real numbers is nonnegative because both terms are positive. Therefore, the roots are always real. To have equal roots, we need (-m-n)^2 + 4p^2 = 0, but this is only true when m = -n and p = 0, which is simply the equation x^2 = 0.