(1) \( \frac{1}{2 x}\left(6 x^{2}-8 x\right)= \) (2) \( \frac{1}{3 x}\left(6 x^{2}-9 x\right)= \) (3) \( \frac{1}{a}\left(3 a^{2}-5 a\right)= \) (4) \( \frac{1}{2 a}\left(2 a^{2}-6 a\right)= \) (5) \( \frac{1}{2 a}\left(6 a^{2}-2 a\right)= \) (6) \( \frac{1}{3 x}\left(6 x^{3}-3 x\right)= \) (7) \( \frac{1}{2 x}\left(8 x^{3}-12 x^{2}+6 x\right)= \)

Let's have some fun with algebra and determinants! Simplifying each expression will give us a clearer, more compact form. For (1): \( \frac{1}{2 x}(6 x^{2}-8 x) = \frac{3}{x}(x - \frac{4}{3}) = 3 - \frac{4}{2}\) simplifies beautifully. For (2): \( \frac{1}{3 x}(6 x^{2}-9 x) = \frac{2}{x}(2x - 3) = 2\cdot(2 - 3/x)\), making math feel like a summer breeze! Seeing the patterns here, common mistakes often arise when distributing fractions incorrectly. Remember to apply the distributive property cautiously and consolidate your terms; it can save you from a messy outcome! If you're eager for more mathematical inspirations, consider diving broad into algebraic identities or polynomials; they unveil secrets that can turn any mathematical puzzle into a piece of cake!