To solve for the missing value, we need to first find a common denominator for the fractions on the left side of the equation. The common denominator is (m - n)(m + n)(m - 1). Thus, we can write:
[(m - n) - ?]/[(m - n)(m + n)(m - 1)] = 2m/(m^2 - n^2)
Next, we can cross-multiply to eliminate the fractions:
[(m - n) - ?](m^2 - n^2) = 2m[(m - n)(m + n)(m - 1)]
Simplifying the right side of the equation:
2m[(m - n)(m + n)(m - 1)] = 2m(m - n)(m + n)(m - 1)
= 2m(m^2 - n^2)(m - 1)
Expanding the left side of the equation:
[(m - n)(m + n) - ?](m^2 - n^2) = (m - n)(m + n)(m - 1)
Distributing the left side of the equation:
(m - n)(m^2 - n^2 + ?) = (m - n)(m + n)(m - 1)
Canceling out the common factor (m - n):
m^2 - n^2 + ? = (m + n)(m - 1)
Expanding the left side of the equation:
? = (m + n)(m - 1) - (m^2 - n^2)
Simplifying the right side of the equation:
? = m^2 - m + n^2 + n - m^2 + n^2
= 2n^2 - m + n
Therefore, the missing value is ? = 2n^2 - m + n.