To simplify \(\frac{6}{x+5} + \frac{4}{x-2}\), we need to find a common denominator.

1. The denominators are \(x+5\) and \(x-2\).
2. The least common denominator (LCD) is \((x+5)(x-2)\).
3. Rewrite each fraction with this common denominator:

\[
\frac{6}{x+5} = \frac{6(x-2)}{(x+5)(x-2)}
\]
\[
\frac{4}{x-2} = \frac{4(x+5)}{(x-2)(x+5)}
\]

4. Now, combine the numerators:

\[
\frac{6(x-2)}{(x+5)(x-2)} + \frac{4(x+5)}{(x+5)(x-2)} = \frac{6(x-2) + 4(x+5)}{(x+5)(x-2)}
\]

5. Expand the numerators:

\[
6(x-2) = 6x - 12 \quad \text{and} \quad 4(x+5) = 4x + 20
\]

So, the expression becomes:

\[
\frac{6x - 12 + 4x + 20}{(x+5)(x-2)} = \frac{10x + 8}{(x+5)(x-2)}
\]

6. Final simplified form:

\[
\frac{10x + 8}{(x+5)(x-2)}
\]

This is the simplified expression.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How would you simplify if the signs of the denominators were reversed?
2. How does this process change if you include negative coefficients in the numerators?
3. Can you simplify \(\frac{5}{x+3} + \frac{7}{x-4}\)?
4. How would this simplification work if one term were squared?
5. What if the variables involved are quadratic, how does the process change?

**Tip:** Always check if the numerator can be factored further after combining terms for possible additional simplification.

Learn how to simplify the expression (6/(x+5)) + (4/(x-2)) by finding a common denominator and combining the numerators. This problem explores key algebra concepts related to rational expressions and is suitable for students in Grades 9-11.

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