https://file.aiquickdraw.com/mathbot/file/1734332958668z7fsfitl.jpg

Let's simplify each part of the expressions step by step:

---

### **A**:  
\[
\left( \frac{a^2 + b^2}{ab} - 1 \right) \cdot \frac{2ab}{a - b}
\]

1. Combine terms in the parenthesis:
\[
\frac{a^2 + b^2}{ab} - 1 = \frac{a^2 + b^2 - ab}{ab}
\]

2. Simplify the entire expression:
\[
\left( \frac{a^2 + b^2 - ab}{ab} \right) \cdot \frac{2ab}{a - b}
\]

- Multiply the terms:
\[
\frac{a^2 + b^2 - ab}{ab} \cdot \frac{2ab}{a - b} = \frac{2(a^2 + b^2 - ab)}{a - b}
\]

---

### **Б**:  
\[
(a - b + \frac{2ab}{a - b}) \cdot \frac{a^2 - b^2}{a^2 + b^2}
\]

1. Combine terms in the parenthesis:
\[
a - b + \frac{2ab}{a - b}
\]

- Find the common denominator:
\[
\frac{(a - b)^2 + 2ab}{a - b} = \frac{a^2 - 2ab + b^2 + 2ab}{a - b} = \frac{a^2 + b^2}{a - b}
\]

2. Multiply with the second term:
\[
\frac{a^2 + b^2}{a - b} \cdot \frac{a^2 - b^2}{a^2 + b^2}
\]

- Simplify:
\[
\frac{a^2 - b^2}{a - b}
\]

- Recognize that \( a^2 - b^2 = (a - b)(a + b) \):
\[
\frac{a^2 - b^2}{a - b} = a + b
\]

---

### **В**:  
\[
\left( \frac{1}{m - n} - \frac{n}{m^2 - mn} \right) : \frac{m}{n - m}
\]

1. Factor \( m^2 - mn \) in the denominator:
\[
m^2 - mn = m(m - n)
\]

Thus:
\[
\frac{n}{m^2 - mn} = \frac{n}{m(m - n)}
\]

2. Simplify the parenthesis:
\[
\frac{1}{m - n} - \frac{n}{m(m - n)} = \frac{m - n - n}{m(m - n)} = \frac{m - n - n}{m(m - n)} = \frac{m - 2n}{m(m - n)}
\]

3. Division by \( \frac{m}{n - m} \):
- Recall \( n - m = -(m - n) \), so:
\[
\frac{m}{n - m} = -\frac{m}{m - n}
\]

- Division becomes multiplication by the reciprocal:
\[
\frac{m - 2n}{m(m - n)} \cdot -\frac{m}{m - n}
\]

4. Combine terms:
\[
-\frac{m(m - 2n)}{m(m - n)^2}
\]

- Cancel \( m \) in the numerator and denominator:
\[
-\frac{m - 2n}{(m - n)^2}
\]

---

### Final Answers:
**A**: \(\frac{2(a^2 + b^2 - ab)}{a - b}\)  
**Б**: \(a + b\)  
**В**: \(-\frac{m - 2n}{(m - n)^2}\)

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Let me know if you want more steps or details!  
### Related Questions:
1. How can we simplify fractions with polynomials?
2. How do you handle complex fractions involving multiple terms?
3. What is the difference between factoring and expanding polynomials?
4. How do we deal with division in algebraic expressions?
5. What is the formula for the difference of squares?

**Tip**: Always factor where possible to simplify algebraic fractions quickly!

Simplify the following algebraic expressions:
A) ( (a^2 + b^2) / ab - 1 ) * (2ab / (a - b))
B) (a - b + (2ab / (a - b))) * ((a^2 - b^2) / (a^2 + b^2))
C) ( (1 / (m - n)) - (n / (m^2 - mn)) ) : (m / (n - m))

Learn how to simplify algebraic expressions involving rational fractions, polynomials, and factorizations. This includes handling differences of squares, combining like terms, and simplifying fractions. Perfect for students in Grades 9-11 studying algebra.

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