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This is incorrect:
Before adding or subtracting rational expressions with different​ denominators, find the LCM of the denominators.  To find the​ LCM, begin by factoring the denominators.

The equation shown in the image is:

\[
\frac{x}{x^2 - 9x + 8} - \frac{x}{x^2 - 4x - 32} = \frac{4}{x + 8}
\]

The accompanying message suggests that the solution is incorrect and that the error occurred because the rational expressions were not properly simplified before being added or subtracted. Specifically, it advises finding the least common multiple (LCM) of the denominators by factoring them.

### Step-by-Step Solution:

1. **Factor the Denominators:**
   - For \( x^2 - 9x + 8 \):
     \[
     x^2 - 9x + 8 = (x - 1)(x - 8)
     \]
   - For \( x^2 - 4x - 32 \):
     \[
     x^2 - 4x - 32 = (x - 8)(x + 4)
     \]
   - The third term \( x + 8 \) is already factored.

2. **Find the LCM of the Denominators:**
   - The LCM of \( (x - 1)(x - 8) \) and \( (x - 8)(x + 4) \) is:
     \[
     \text{LCM} = (x - 1)(x - 8)(x + 4)
     \]

3. **Rewrite Each Term with the LCM as the Denominator:**
   - First term:
     \[
     \frac{x}{(x - 1)(x - 8)} \rightarrow \frac{x \cdot (x + 4)}{(x - 1)(x - 8)(x + 4)}
     \]
   - Second term:
     \[
     \frac{x}{(x - 8)(x + 4)} \rightarrow \frac{x \cdot (x - 1)}{(x - 1)(x - 8)(x + 4)}
     \]
   - Third term:
     \[
     \frac{4}{x + 8} \rightarrow \frac{4 \cdot (x - 1)(x - 8)}{(x - 1)(x - 8)(x + 4)}
     \]

4. **Combine the Terms:**
   - Combine the numerators over the common denominator:
     \[
     \frac{x(x + 4) - x(x - 1)}{(x - 1)(x - 8)(x + 4)} = \frac{4(x - 1)(x - 8)}{(x - 1)(x - 8)(x + 4)}
     \]
   - Simplify the numerators on both sides and compare.

### Conclusion:
The original equation did not account for the proper LCM of the denominators, leading to an incorrect equation. By factoring the denominators and finding the LCM, you ensure all terms have a common denominator before combining them, which is critical for accurate results.

Do you need further details or clarification? 

### Follow-Up Questions:
1. What happens when you simplify the left-hand side of the equation?
2. How do you factor trinomials like the ones in this problem?
3. What is the general process for finding the LCM of polynomial expressions?
4. How would the equation change if a constant were added to both sides?
5. How do you check your final answer in a problem like this?

**Tip:** Always factor the denominators fully before performing operations with rational expressions to ensure accuracy.

Learn how to correct rational expressions with different denominators by finding the LCM and factoring the denominators. This detailed step-by-step solution covers key algebraic concepts and ensures accuracy in solving equations involving rational expressions. Suitable for high school students.

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