Divide. \[ \frac{x^{2}+x-2}{x-2} \div \frac{3 x+6}{x-3} \] Simplify your answer as much as possible. \( \square \)

\[ \frac{x^{2}+x-2}{x-2} \div \frac{3x+6}{x-3} \] We start by rewriting the division as a multiplication by the reciprocal: \[ \frac{x^{2}+x-2}{x-2} \cdot \frac{x-3}{3x+6} \] Factor the quadratic \(x^{2}+x-2\): \[ x^{2}+x-2 = (x+2)(x-1) \] Factor the expression \(3x+6\): \[ 3x+6 = 3(x+2) \] Substitute these factors back into the expression: \[ \frac{(x+2)(x-1)}{x-2} \cdot \frac{x-3}{3(x+2)} \] Cancel the common factor \((x+2)\) (assuming \(x \neq -2\)): \[ \frac{x-1}{x-2} \cdot \frac{x-3}{3} \] Multiply the numerators and denominators: \[ \frac{(x-1)(x-3)}{3(x-2)} \] Thus, the simplified expression is: \[ \boxed{\frac{(x-1)(x-3)}{3(x-2)}} \] Note that in the original expression there are some restrictions on \(x\): \(x \neq 2\) (from the denominator \(x-2\)); \(x \neq -2\) (from the canceled factor \(x+2\)); and \(x \neq 3\) (from the denominator \(x-3\) in the division).