The expression you provided is:

\[
\frac{3x}{x+2} + \frac{x}{x-2}
\]

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

Let's rewrite both fractions with the common denominator:

\[
\frac{3x}{x+2} = \frac{3x(x-2)}{(x+2)(x-2)}
\]
\[
\frac{x}{x-2} = \frac{x(x+2)}{(x+2)(x-2)}
\]

Now we can add the two fractions:

\[
\frac{3x(x-2)}{(x+2)(x-2)} + \frac{x(x+2)}{(x+2)(x-2)} = \frac{3x(x-2) + x(x+2)}{(x+2)(x-2)}
\]

Simplify the numerators:

\[
3x(x-2) = 3x^2 - 6x
\]
\[
x(x+2) = x^2 + 2x
\]

Now substitute these into the expression:

\[
\frac{3x^2 - 6x + x^2 + 2x}{(x+2)(x-2)}
\]

Combine like terms:

\[
\frac{3x^2 + x^2 - 6x + 2x}{(x+2)(x-2)} = \frac{4x^2 - 4x}{(x+2)(x-2)}
\]

Factor the numerator:

\[
\frac{4x(x-1)}{(x+2)(x-2)}
\]

Thus, the simplified expression is:

\[
\frac{4x(x-1)}{(x+2)(x-2)}
\]

Let me know if you'd like further clarification!

### Related Questions:
1. How do you find the least common denominator for rational expressions?
2. What are the steps to simplify rational expressions?
3. Can you factor more complex quadratic expressions?
4. How do you simplify expressions involving multiple variables?
5. How do restrictions on variable values affect rational expressions?

**Tip**: Always check for values that would make the denominator zero—these are excluded from the domain!

Discover how to simplify the rational expression 3x/(x+2) + x/(x-2) by finding the least common denominator, rewriting fractions, and factoring quadratic expressions. This guide is ideal for students in Grades 9-11 studying algebra.

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