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.

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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 $$\frac{x-6}{-42}$$.
Alicia made mistakes in factoring and canceling common terms. The correct multiplication result is $$ \frac{x}{6} $$.

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