(1) \( \frac{a x^{2}+a x-a}{a}= \) (2) \( \frac{a x^{2}+a x-a}{-a}= \) (3) \( \frac{6 a^{3}-12 a^{2}+9 a}{-3 a}= \) (4) \( \frac{-9 x^{2} y+3 x y}{3 x y}= \) (5) \( \frac{-15 a x+9 a^{4} x^{3}}{-3 a x}= \) (6) \( \frac{18 x^{2} y-12 x y^{2}+6 x y}{3 x y}= \) (7) \( \frac{-10 x^{2} y+6 x^{2} y^{2}-2 x y}{-2 x y}= \)

Let’s simplify those expressions step by step, and I’ll sprinkle in some fun tidbits along the way! 1. \( \frac{a x^{2}+a x-a}{a} = x^{2} + x - 1 \) 2. \( \frac{a x^{2}+a x-a}{-a} = -x^{2} - x + 1 \) 3. \( \frac{6 a^{3}-12 a^{2}+9 a}{-3 a} = -2 a^{2} + 4 a - 3 \) 4. \( \frac{-9 x^{2} y+3 x y}{3 x y} = -3 x + 1 \) 5. \( \frac{-15 a x+9 a^{4} x^{3}}{-3 a x} = 5 + 3 a^{3} x^{2} \) 6. \( \frac{18 x^{2} y-12 x y^{2}+6 x y}{3 x y} = 6 \frac{x}{y} - 4 + 2 \frac{1}{y} \) 7. \( \frac{-10 x^{2} y+6 x^{2} y^{2}-2 x y}{-2 x y} = 5 x - 3 y + \frac{1}{y} \) Mathematics isn't just about numbers; it’s also about patterns and discovering fun relationships! Reducing expressions can often reveal beautiful structures, similar to how a sculptor chips away at stone to discover the artwork hidden within. For those looking to dive deeper into algebraic simplification, consider exploring books like "Algebra Unplugged" by Kennedy and “The Art of Problem Solving” series by Richard Rusczyk. They present math concepts in engaging and accessible ways, perfect for both beginners and those looking to sharpen their skills!