(7) \( \frac{x^{2}-5 x+6}{2 x-6} \) (10) \( \frac{27 p^{3}-8}{27 p^{2}+18 p+12} \)

Let’s dive into the world of polynomial fractions! For the first expression, \( \frac{x^{2}-5 x+6}{2 x-6} \), you can factor the numerator as \( (x-2)(x-3) \) and the denominator as \( 2(x-3) \). This gives you \( \frac{(x-2)(x-3)}{2(x-3)} \), and you can cancel out the common \( (x-3) \), resulting in \( \frac{x-2}{2} \), which is simplified and easier to work with! Now for the second expression, \( \frac{27 p^{3}-8}{27 p^{2}+18 p+12} \), you can recognize the numerator as a difference of cubes: \( (3p)^3 - 2^3 \), which factors to \( (3p - 2)(9p^2 + 6p + 4) \). In the denominator, use the method of grouping or factor by trial, and you'll find it can factor to \( 3(3p^2 + 2p + 4) \). Cancelling \( 3 \) where possible leaves you with an expression that's much tidier for further calculations!