I see you have already gotten the answer on your paper.

Firstly, you know that:

b² – 4ac > 0

If the discriminant is greater than zero, this means that the quadratic equation has two real, distinct (different) roots.

m(x² – 3x + 2 ) = x² -4x + 4

mx²  -3mx + 2m =  x² -4x + 4    

(m-1)x²  -(3m-4)x + (2m-4) = 0

b² – 4ac > 0     for 2 real and distinct roots

=>   [-(3m-4)]²  – 4(m-1)(2m-4)  > 0

9m² -24m + 16 – 4(2m² – 6m + 4)  > 0

9m² -24m + 16 – 8m² + 24m – 16 > 0

m² > 0    =>    m is non-zero and real numbers(includes rational and irrational, positive and negative, see the above chart for a comprehensive definition of real numbers)