3. Simplify. a) \( \frac{a b-a^{2}}{b^{2}-a^{2}} \div \frac{a^{2}}{b^{2}+a b} \) (b) \( \frac{3 x^{2}+27}{2 x+6} \times \frac{6 x-18}{x^{4}-81} \) c) \( \frac{2 x^{2}-6 x}{x^{2}-1} \div \frac{x-3}{x^{2}-x} \div \frac{1}{3 x+3} \) (d) \( \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \frac{x^{3}-2 x^{2}}{x^{2}-16} \times \frac{1}{x+4} \) (f) \( \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} \) (d) \( \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \)

To simplify these expressions, let's make use of factorization and algebraic identities, making sure we keep everything neat and tidy! First Line: It's crucial to remember that when you encounter a division of fractions, it can be transformed into multiplication by flipping the second fraction. For instance, \( \frac{a}{b} \div \frac{c}{d} \) becomes \( \frac{a}{b} \times \frac{d}{c} \). This simple step can lead to easier simplifications, especially when dealing with polynomials that can be factored. Second Line: Look out for common factors in the numerators and denominators! For example, in expressions like \( \frac{x^2 - 9}{x^2 - 6x + 9} \), recognizing that \( x^2 - 9 \) factors to \( (x + 3)(x - 3) \) and \( x^2 - 6x + 9 \) is \( (x - 3)^2 \) opens the door to canceling terms. Always factor first, it’s like wielding a magic wand of simplification!