https://file.aiquickdraw.com/mathbot/file/1726505707076fr6jlrrq.jpg

The expression given in the image is:

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

Let's break this down step by step.

### Step 1: Simplify the numerator
The numerator is:
\[
x + \frac{2x}{x - 2}
\]
To simplify this, we need a common denominator:
\[
\frac{x(x - 2)}{x - 2} + \frac{2x}{x - 2} = \frac{x(x - 2) + 2x}{x - 2}
\]
Expanding \(x(x - 2)\):
\[
x(x - 2) = x^2 - 2x
\]
Thus, the numerator becomes:
\[
\frac{x^2 - 2x + 2x}{x - 2} = \frac{x^2}{x - 2}
\]

### Step 2: Simplify the denominator
The denominator is:
\[
1 + \frac{4}{(x + 2)(x - 2)}
\]
This can be rewritten with a common denominator:
\[
\frac{(x + 2)(x - 2)}{(x + 2)(x - 2)} + \frac{4}{(x + 2)(x - 2)} = \frac{(x + 2)(x - 2) + 4}{(x + 2)(x - 2)}
\]
Expanding \((x + 2)(x - 2)\):
\[
(x + 2)(x - 2) = x^2 - 4
\]
Thus, the denominator becomes:
\[
\frac{x^2 - 4 + 4}{x^2 - 4} = \frac{x^2}{x^2 - 4}
\]

### Step 3: Combine the numerator and denominator
We now have:
\[
\frac{\frac{x^2}{x - 2}}{\frac{x^2}{x^2 - 4}} = \frac{x^2}{x - 2} \times \frac{x^2 - 4}{x^2}
\]
Cancel out \(x^2\) in the numerator and denominator:
\[
\frac{x^2 - 4}{x - 2}
\]
Now, recall that:
\[
x^2 - 4 = (x - 2)(x + 2)
\]
So the expression becomes:
\[
\frac{(x - 2)(x + 2)}{x - 2}
\]
Finally, cancel out \(x - 2\) (as long as \(x \neq 2\)):
\[
x + 2
\]

Thus, the simplified expression is:

\[
x + 2
\]

Would you like further explanation or details? Here are some follow-up questions:

1. How do common denominators work when simplifying complex fractions?
2. What conditions must be considered when canceling out terms like \((x - 2)\)?
3. How does factoring help in simplifying rational expressions?
4. Can you solve this type of problem using another method?
5. What are the restrictions on the variable \(x\) in the original expression?

**Tip:** Always check for restrictions on the variable when simplifying rational expressions (like \(x \neq 2\) and \(x \neq -2\) here).

Simplify the expression: (x + (2x / (x - 2))) / (1 + (4 / ((x + 2)(x - 2))))

Learn how to simplify the rational expression (x + (2x / (x - 2))) / (1 + (4 / ((x + 2)(x - 2)))) by applying common denominators, factoring, and cancellation techniques. This problem demonstrates key algebraic concepts, suitable for students in Grades 10-12.

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