It is about the discriminant of the quadratic equation.
The discriminant in this case is d = b^2 - 4ac,  where

a = 1, b = (m-2), and c = 4.

So, d = (m-2)^2 - 4*4 = .


A)  The equation  =   has exactly 2 real solutions if and only if d > 0.

    It is equivalent to an inequality   > , which has the solutions  m < -2  and/or  m > 6.



B)  The equation  =   has exactly 1 real solutions if and only if d = 0.

   It is equivalent to an equation   = ,  which has two solutions m = 6  and/or  m = -2.


C)  The equation  =   has exactly 0 real solutions if and only if d < 0.

    It is equivalent to an inequality   < , which has the solutions  -2 < m < 6.


Answer. A) has exactly 2 real solutions  if and only if  m < -2  and/or  m > 6.
        B) has exactly 1 real solution   if and only if  m = -2  and/or  m = 6.
        C) has exactly 0 real solutions  if and only if  -2 < m < 6.