2. Alicia multiplied \( \frac{3 x-6}{8 x-16} \cdot \frac{4 x}{9} \). Her work is shown. Alicia \[ \begin{aligned} \frac{3 x-6}{8 x-16} \cdot \frac{4 x}{9}= & \frac{3^{\prime} x-6}{8 x-16} \cdot \frac{4^{\prime} x}{9} \\ & =\frac{x-6}{-14} \cdot \frac{1}{3} \\ & =\frac{x-6}{-42} ; x \neq 2 \end{aligned} \] Identify the error(s) in Alicia's work. Then correctly perform the multiplication.

Let's analyze Alicia's work step by step to identify the errors and then correctly perform the multiplication. ### Step 1: Identify Errors in Alicia's Work 1. **Factoring Errors**: - Alicia factored \(3x - 6\) incorrectly. The correct factorization is: \[ 3x - 6 = 3(x - 2) \] - She also factored \(8x - 16\) incorrectly. The correct factorization is: \[ 8x - 16 = 8(x - 2) \] 2. **Multiplication Errors**: - After factoring, Alicia wrote: \[ \frac{3^{\prime} x - 6}{8 x - 16} \cdot \frac{4^{\prime} x}{9} \] This notation is incorrect. The correct expression after factoring should be: \[ \frac{3(x - 2)}{8(x - 2)} \cdot \frac{4x}{9} \] 3. **Cancellation Errors**: - Alicia did not cancel the common factor \(x - 2\) correctly. The \(x - 2\) terms in the numerator and denominator can be canceled. ### Step 2: Correctly Perform the Multiplication Now, let's perform the multiplication correctly: 1. **Write the Expression**: \[ \frac{3(x - 2)}{8(x - 2)} \cdot \frac{4x}{9} \] 2. **Cancel Common Factors**: - The \(x - 2\) terms cancel out: \[ = \frac{3}{8} \cdot \frac{4x}{9} \] 3. **Multiply the Remaining Terms**: \[ = \frac{3 \cdot 4x}{8 \cdot 9} = \frac{12x}{72} \] 4. **Simplify the Fraction**: \[ = \frac{x}{6} \] ### Final Result The correct result of the multiplication is: \[ \frac{x}{6}, \quad x \neq 2 \] ### Summary of Errors - Incorrect factorization of \(3x - 6\) and \(8x - 16\). - Incorrect cancellation of common factors. - The final answer should be \(\frac{x}{6}\) instead of \(\frac{x-6}{-42}\).