Multiply. \[ \frac{x+2}{x^{2}-2 x-3} \cdot \frac{x^{2}-4 x+3}{3 x+6} \] Simplify your answer as much as possible.

To simplify the expression \[ \frac{x+2}{x^{2}-2x-3} \cdot \frac{x^{2}-4x+3}{3x+6}, \] let's begin by factoring each part of the expression. 1. Factor \( x^2 - 2x - 3 \): \[ x^2 - 2x - 3 = (x - 3)(x + 1). \] 2. Factor \( x^2 - 4x + 3 \): \[ x^2 - 4x + 3 = (x - 3)(x - 1). \] 3. Factor \( 3x + 6 \): \[ 3x + 6 = 3(x + 2). \] Now we can rewrite the original expression using these factorizations: \[ \frac{x + 2}{(x - 3)(x + 1)} \cdot \frac{(x - 3)(x - 1)}{3(x + 2)}. \] Next, we can simplify this expression. Cancellation can occur with \( x + 2 \) and \( x - 3 \): \[ = \frac{(x + 2)(x - 3)(x - 1)}{(x - 3)(x + 1)(3(x + 2))}. \] Cancel the common factors \( (x - 3) \) and \( (x + 2) \): \[ = \frac{(x - 1)}{3(x + 1)}. \] After cancellation, we obtain: \[ \frac{x - 1}{3(x + 1)}. \] So the final simplified expression is: \[ \frac{x - 1}{3(x + 1)}. \]