(2) Divide and state the quotient in simplest form. \( \frac{x^{2}-9 x+18}{x+1} \div \frac{x^{2}+x-12}{x+1} \) \( \frac{x+4}{x-6} \) \( \frac{x^{2}-9 x+18}{x^{2}+x-12} \) \( \frac{x-6}{x+4} \) \( \frac{-3}{2} \)

To divide the two rational expressions, you can multiply the first fraction by the reciprocal of the second fraction. This means we switch the second fraction and change the division into multiplication. So, you'll start with: \[ \frac{x^{2}-9x+18}{x+1} \times \frac{x+1}{x^{2}+x-12} \] The \(x+1\) terms in the numerator and denominator cancel out, simplifying our expression to: \[ \frac{x^{2} - 9x + 18}{x^{2} + x - 12} \] Next, factor both polynomials. The numerator factors to \((x-6)(x-3)\) and the denominator factors to \((x-3)(x+4)\). You can cancel out the \((x-3)\) term, leading to: \[ \frac{x-6}{x+4} \] Thus, the quotient in simplest form is: \[ \frac{x-6}{x+4} \] Applying this process helps grasp the beautiful symmetry in algebra as you can see common factors simplifying your expressions! But remember, always check for restrictions! The original denominators can't equal zero, giving you important conditions to watch out for: \(x \neq -1\), \(x \neq 3\), and \(x \neq -4\).