Divide. \[ \frac{x^{2}-5 x+6}{x-1} \div \frac{3 x-9}{x+2} \] Simplify your answer as much as possible.

To simplify the expression \[ \frac{x^{2}-5 x+6}{x-1} \div \frac{3 x-9}{x+2}, \] we start by rewriting the division as multiplication by the reciprocal: \[ \frac{x^{2}-5 x+6}{x-1} \times \frac{x+2}{3 x-9}. \] Next, let's factor the polynomials in the expression. The quadratic \( x^2 - 5x + 6 \) can be factored as: \[ x^2 - 5x + 6 = (x-2)(x-3). \] Now, let's factor \( 3x - 9 \): \[ 3x - 9 = 3(x-3). \] Now substituting these factorizations back into the expression, we have: \[ \frac{(x-2)(x-3)}{x-1} \times \frac{x+2}{3(x-3)}. \] Next, we can cancel out the common factors. The \( (x-3) \) terms in the numerator and denominator cancel: \[ \frac{(x-2)(\cancel{x-3})}{x-1} \times \frac{x+2}{3\cancel{(x-3)}} = \frac{(x-2)(x+2)}{3(x-1)}. \] Now we have a simplified expression: \[ \frac{(x-2)(x+2)}{3(x-1)}. \] We can further expand the numerator: \[ (x-2)(x+2) = x^2 - 4. \] Thus, the final simplified expression is: \[ \frac{x^2 - 4}{3(x-1)}. \] Therefore, the answer is: \[ \frac{x^2 - 4}{3(x-1)}. \]