1. **State the problem:** Simplify the expression $$23. \left( \frac{a + b}{a^2 + b^2} - \frac{5a + b}{a^2 - b^2} : \frac{5a^3 + a^2b + 5ab^2 + b^3}{2ab(a - b)} \right) \cdot \left( \frac{a^2 - b^2}{ab} + \frac{1}{a} \right)$$ 2. **Rewrite the division as multiplication by reciprocal:** $$\frac{a + b}{a^2 + b^2} - \frac{5a + b}{a^2 - b^2} \div \frac{5a^3 + a^2b + 5ab^2 + b^3}{2ab(a - b)} = \frac{a + b}{a^2 + b^2} - \frac{5a + b}{a^2 - b^2} \times \frac{2ab(a - b)}{5a^3 + a^2b + 5ab^2 + b^3}$$ 3. **Factor where possible:** - $a^2 - b^2 = (a - b)(a + b)$ - Factor numerator $5a^3 + a^2b + 5ab^2 + b^3$ by grouping: $$5a^3 + a^2b + 5ab^2 + b^3 = a^2(5a + b) + b^2(5a + b) = (a^2 + b^2)(5a + b)$$ 4. **Substitute factorizations:** $$\frac{a + b}{a^2 + b^2} - \frac{5a + b}{(a - b)(a + b)} \times \frac{2ab(a - b)}{(a^2 + b^2)(5a + b)}$$ 5. **Simplify the multiplication:** $$\frac{5a + b}{(a - b)(a + b)} \times \frac{2ab(a - b)}{(a^2 + b^2)(5a + b)} = \frac{2ab}{a^2 + b^2}$$ 6. **Rewrite the expression inside the parentheses:** $$\frac{a + b}{a^2 + b^2} - \frac{2ab}{a^2 + b^2} = \frac{a + b - 2ab}{a^2 + b^2}$$ 7. **Simplify the numerator:** No further factorization possible, keep as is. 8. **Simplify the second parentheses:** $$\frac{a^2 - b^2}{ab} + \frac{1}{a} = \frac{(a - b)(a + b)}{ab} + \frac{1}{a} = \frac{(a - b)(a + b)}{ab} + \frac{b}{ab} = \frac{(a - b)(a + b) + b}{ab}$$ 9. **Expand numerator:** $$(a - b)(a + b) + b = (a^2 - b^2) + b = a^2 - b^2 + b$$ 10. **Final expression:** $$\left( \frac{a + b - 2ab}{a^2 + b^2} \right) \times \left( \frac{a^2 - b^2 + b}{ab} \right)$$ This is the simplified form. **Final answer:** $$\frac{(a + b - 2ab)(a^2 - b^2 + b)}{ab(a^2 + b^2)}$$