(m+2)/(2) - (m)/(m+2)
+(2m^2-m^3)/(2m^2-8)
2m^2-8 = 2(m-2)(m+2)
so, that is the common denominator. You end up with
(m+2)(m-2)(m+2) - m(2)(m-2) + 2m^2-m^3
----------------------------------------
2(m-2)(m+2)
m^3+2m^2-4m-8 - (2m^2-4m) + (2m^2-m^3)
----------------------------------------
2(m-2)(m+2)
2m^2-8
------------
2m^2-8
1