(2) \( \left(\frac{2}{m^{2}-m}-\frac{2 m}{1-m^{2}}\right) \cdot \frac{2 m^{2}+2 m}{m^{3}-1}+\frac{4}{m-1}= \)

We wish to simplify   [2/(m² – m) – 2m/(1 – m²)] · [(2m² + 2m)/(m³ – 1)] + 4/(m – 1). Follow these steps: 1. Write the first bracket in a common form.  a. Notice that    m² – m = m(m – 1).  b. Also, note that    1 – m² = –(m² – 1) = –[(m – 1)(m + 1)].  c. Thus, we have    2/(m² – m) = 2/[m(m – 1)]    and    2m/(1 – m²) = 2m/[-(m – 1)(m + 1)] = –2m/[(m – 1)(m + 1)].  d. So the first bracket becomes:    2/[m(m – 1)] – [–2m/((m – 1)(m + 1))] = 2/[m(m – 1)] + 2m/[(m – 1)(m + 1)]. 2. Factor out 2/(m – 1):   2/[m(m – 1)] + 2m/[(m – 1)(m + 1)] = 2/(m – 1) · [1/m + m/(m + 1)].  Combine the terms in the bracket:   1/m + m/(m + 1) = ( (m + 1) + m² )/[m(m + 1)] = (m² + m + 1)/[m(m + 1)].  So the first bracket simplifies to:   2(m² + m + 1)/[m(m + 1)(m – 1)]. 3. Simplify the second factor.  a. Write the numerator:   2m² + 2m = 2m(m + 1).  b. Factor the denominator. Since m³ – 1 is a difference of cubes,   m³ – 1 = (m – 1)(m² + m + 1).  c. Therefore, the second factor becomes:   [2m(m + 1)]/[(m – 1)(m² + m + 1)]. 4. Multiply the two simplified factors:  We have   [2(m² + m + 1)/{m(m + 1)(m – 1)}] · [2m(m + 1)/{(m – 1)(m² + m + 1)}].  Multiply the numerators and the denominators:   Numerator: 2·2·m·(m + 1)·(m² + m + 1) = 4m(m + 1)(m² + m + 1),   Denominator: m(m + 1)(m – 1) · (m – 1)(m² + m + 1) = m(m + 1)(m² + m + 1)(m – 1)².  Cancel the common factors m, (m + 1) and (m² + m + 1):   Result = 4/(m – 1)². 5. Now include the remaining term:  The whole expression is now   4/(m – 1)² + 4/(m – 1).  To combine these fractions use the common denominator (m – 1)²:   = [4 + 4(m – 1)]/(m – 1)²   = [4 + 4m – 4]/(m – 1)²   = 4m/(m – 1)². Therefore, the given expression simplifies to:   4m/(m – 1)².