https://file.aiquickdraw.com/mathbot/file/1730175807209l6gi6efu.jpg

The problem shown in the image is:

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

To add these two fractions, we need to find a common denominator. The denominators here are \(x + 2\) and \(x - 4\), so the common denominator is \((x + 2)(x - 4)\).

Let's rewrite each fraction with this common denominator:

1. Rewrite \(\frac{6a}{x+2}\) as \(\frac{6a(x - 4)}{(x+2)(x-4)}\).
2. Rewrite \(\frac{2a}{x-4}\) as \(\frac{2a(x + 2)}{(x+2)(x-4)}\).

Now combine the two fractions:

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

Next, expand the numerators:

\[
= \frac{6a \cdot x - 6a \cdot 4 + 2a \cdot x + 2a \cdot 2}{(x+2)(x-4)}
\]
\[
= \frac{6ax - 24a + 2ax + 4a}{(x+2)(x-4)}
\]

Combine like terms:

\[
= \frac{(6ax + 2ax) - 24a + 4a}{(x+2)(x-4)}
\]
\[
= \frac{8ax - 20a}{(x+2)(x-4)}
\]

So, the simplified expression is:

\[
\frac{8ax - 20a}{(x+2)(x-4)}
\]

Would you like further details or have any questions about this?

### Additional Questions
1. How can you simplify expressions with different denominators?
2. What are the steps for factoring expressions in the numerator and denominator?
3. How do you handle complex fractions involving multiple terms?
4. What is the importance of non-zero denominators in rational expressions?
5. How do you find the least common multiple of polynomial denominators?

### Tip
Always factor out any common terms in both the numerator and denominator to check for further simplification.

Simplify the expression (6a / (x + 2)) + (2a / (x - 4)). Assume all denominators are non-zero.

Learn how to simplify the rational expression (6a / (x + 2)) + (2a / (x - 4)). This solution covers the concept of finding a common denominator and combining terms in rational expressions. Suitable for students in Grades 9-11.

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