1. **State the problem:** Simplify the expression $$\frac{a + b}{a^2 + b^2} - \frac{5a + b}{a^2 - b^2} : \frac{5a^3 + a^2b + 5ab^2 + b^3}{2ab(a - b)}$$ 2. **Recall important formulas and rules:** - Difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ - Division of fractions: $$\frac{A}{B} : \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$$ - Factorization and simplification are key. 3. **Rewrite the division as multiplication:** $$\left(\frac{a + b}{a^2 + b^2} - \frac{5a + b}{a^2 - b^2}\right) \times \frac{2ab(a - b)}{5a^3 + a^2b + 5ab^2 + b^3}$$ 4. **Factor denominators and numerator where possible:** - $$a^2 - b^2 = (a - b)(a + b)$$ - Factor numerator $$5a^3 + a^2b + 5ab^2 + b^3$$ by grouping: $$5a^3 + a^2b + 5ab^2 + b^3 = a^2(5a + b) + b^2(5a + b) = (a^2 + b^2)(5a + b)$$ 5. **Rewrite the expression with factored terms:** $$\left(\frac{a + b}{a^2 + b^2} - \frac{5a + b}{(a - b)(a + b)}\right) \times \frac{2ab(a - b)}{(a^2 + b^2)(5a + b)}$$ 6. **Find common denominator for the subtraction inside parentheses:** Common denominator is $$(a^2 + b^2)(a - b)(a + b)$$ but note $a^2 + b^2$$ and $$(a - b)(a + b) = a^2 - b^2$$ are different, so keep as is. Rewrite subtraction: $$\frac{(a + b)(a - b)(a + b)}{(a^2 + b^2)(a - b)(a + b)} - \frac{(5a + b)(a^2 + b^2)}{(a^2 + b^2)(a - b)(a + b)}$$ But this is complicated; better to write as: $$\frac{(a + b)(a - b)(a + b) - (5a + b)(a^2 + b^2)}{(a^2 + b^2)(a - b)(a + b)}$$ 7. **Simplify numerator of subtraction:** - $$(a + b)(a - b)(a + b) = (a + b)^2 (a - b)$$ - Expand $$(a + b)^2 = a^2 + 2ab + b^2$$ - So numerator is: $$ (a^2 + 2ab + b^2)(a - b) - (5a + b)(a^2 + b^2) $$ 8. **Expand both parts:** - First part: $$ (a^2 + 2ab + b^2)(a - b) = a^3 - a^2b + 2a^2b - 2ab^2 + ab^2 - b^3 = a^3 + a^2b - ab^2 - b^3 $$ - Second part: $$ (5a + b)(a^2 + b^2) = 5a^3 + 5ab^2 + a^2b + b^3 $$ 9. **Subtract second part from first:** $$ (a^3 + a^2b - ab^2 - b^3) - (5a^3 + 5ab^2 + a^2b + b^3) = a^3 - 5a^3 + a^2b - a^2b - ab^2 - 5ab^2 - b^3 - b^3 = -4a^3 - 6ab^2 - 2b^3 $$ 10. **Rewrite the entire expression:** $$ \frac{-4a^3 - 6ab^2 - 2b^3}{(a^2 + b^2)(a - b)(a + b)} \times \frac{2ab(a - b)}{(a^2 + b^2)(5a + b)} $$ 11. **Cancel common factors:** - Cancel $$(a - b)$$ - Expression becomes: $$ \frac{-4a^3 - 6ab^2 - 2b^3}{(a^2 + b^2)(a + b)} \times \frac{2ab}{(a^2 + b^2)(5a + b)} $$ 12. **Factor numerator $$-4a^3 - 6ab^2 - 2b^3$$:** Factor out $$-2$$: $$ -2(2a^3 + 3ab^2 + b^3) $$ 13. **Final expression:** $$ \frac{-2(2a^3 + 3ab^2 + b^3) \times 2ab}{(a^2 + b^2)^2 (a + b)(5a + b)} = \frac{-4ab(2a^3 + 3ab^2 + b^3)}{(a^2 + b^2)^2 (a + b)(5a + b)} $$ This is the simplified form. **Final answer:** $$\boxed{\frac{-4ab(2a^3 + 3ab^2 + b^3)}{(a^2 + b^2)^2 (a + b)(5a + b)}}$$