\( \frac{x^{2}-5 x+6}{x^{2}+4 x+3} \cdot \frac{3 x+9}{x-3} \) Simplify your answer as much as possible.

To simplify the expression \( \frac{x^{2}-5 x+6}{x^{2}+4 x+3} \cdot \frac{3 x+9}{x-3} \), we start by factoring each part where possible. 1. **Factor the numerators and denominators**: - The numerator \( x^2 - 5x + 6 \) factors to \( (x - 2)(x - 3) \). - The denominator \( x^2 + 4x + 3 \) factors to \( (x + 1)(x + 3) \). - The numerator \( 3x + 9 \) can be factored as \( 3(x + 3) \). Now, we substitute our factored forms back into the expression: \[ \frac{(x - 2)(x - 3)}{(x + 1)(x + 3)} \cdot \frac{3(x + 3)}{x - 3} \] 2. **Cancel out common factors**: - We see that \( x - 3 \) in the numerator of the first fraction cancels with \( x - 3 \) in the denominator of the second fraction. - Also, \( (x + 3) \) from the numerator of the second fraction cancels with \( (x + 3) \) in the denominator of the first fraction. The expression simplifies to: \[ \frac{3(x - 2)}{(x + 1)} \] Hence, the simplified form of the given expression is: \[ \frac{3(x - 2)}{x + 1} \] This result holds provided that \( x \neq -3 \), \( x \neq -1 \), and \( x \neq 3 \) to avoid division by zero.