https://file.aiquickdraw.com/mathbot/file/1730127701985zr3ywipn.jpg

The expression provided in the image is:

\[
\frac{6a}{x + 2} + \frac{2a}{x - 4}
\]

with a note specifying that "all denominators are non-zero."

### Steps to Solve or Simplify
1. **Identify Non-Zero Conditions:** 
   For this expression to be defined, neither denominator can be zero. Therefore:
   - \( x + 2 \neq 0 \Rightarrow x \neq -2 \)
   - \( x - 4 \neq 0 \Rightarrow x \neq 4 \)
   
   Thus, \( x \) must not equal \(-2\) or \(4\).

2. **Find a Common Denominator (Optional for Simplification):**
   If you wish to add these two fractions, the common denominator would be \((x + 2)(x - 4)\). This would lead to:
   
   \[
   \frac{6a(x - 4) + 2a(x + 2)}{(x + 2)(x - 4)}
   \]

   Simplifying further would require expanding and combining like terms.

Would you like a detailed simplification of the expression, or do you have any specific question regarding this setup?

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### Related Questions
1. What would be the expression after fully simplifying \(\frac{6a}{x+2} + \frac{2a}{x-4}\)?
2. How would we find the values of \( x \) where this expression equals zero?
3. Could this type of expression be graphed, and if so, what would the graph look like near the undefined points?
4. How would adding another fraction with a different denominator impact the solution process?
5. Can this expression be generalized for other values or parameters?

### Tip
When dealing with fractions, always remember to check and exclude values that make the denominator zero, as they create undefined points in the expression.

Given the expression 6a/(x + 2) + 2a/(x - 4) with a note specifying that all denominators are non-zero, what are the steps to simplify or solve it?

Explore the steps to simplify the rational expression 6a/(x+2) + 2a/(x-4) while ensuring all denominators are non-zero. This problem covers key algebraic concepts such as finding a common denominator and addressing non-zero constraints in rational expressions, making it suitable for high school students in Grades 9-11.

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