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The expression you provided is:

\[
\frac{1 + \frac{3}{w - 2}}{1 - \frac{3}{w + 2}}
\]

Let’s simplify this step by step.

### Step 1: Simplify both the numerator and the denominator.
The numerator is:
\[
1 + \frac{3}{w - 2}
\]
We need to find a common denominator:
\[
\frac{(w - 2)}{(w - 2)} + \frac{3}{w - 2} = \frac{w - 2 + 3}{w - 2} = \frac{w + 1}{w - 2}
\]
So, the simplified numerator is:
\[
\frac{w + 1}{w - 2}
\]

The denominator is:
\[
1 - \frac{3}{w + 2}
\]
Again, find a common denominator:
\[
\frac{(w + 2)}{(w + 2)} - \frac{3}{w + 2} = \frac{w + 2 - 3}{w + 2} = \frac{w - 1}{w + 2}
\]
So, the simplified denominator is:
\[
\frac{w - 1}{w + 2}
\]

### Step 2: Combine the simplified numerator and denominator into one fraction.
Now, we have:
\[
\frac{\frac{w + 1}{w - 2}}{\frac{w - 1}{w + 2}}
\]
When dividing by a fraction, multiply by the reciprocal:
\[
\frac{w + 1}{w - 2} \times \frac{w + 2}{w - 1} = \frac{(w + 1)(w + 2)}{(w - 2)(w - 1)}
\]

### Step 3: Final result.
The simplified expression is:
\[
\frac{(w + 1)(w + 2)}{(w - 2)(w - 1)}
\]

Let me know if you need further clarifications!

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Here are 5 related questions to enhance your understanding:
1. How can we simplify more complex algebraic fractions?
2. What techniques are useful for finding common denominators in rational expressions?
3. How do we factorize and cancel terms when simplifying rational expressions?
4. What is the role of reciprocal multiplication when dividing fractions?
5. Can we apply the same techniques to complex rational expressions involving polynomials?

**Tip**: Always check for possible common factors that can be canceled after simplifying algebraic expressions.

Simplify \( \frac{1 + \frac{3}{w - 2}}{1 - \frac{3}{w + 2}} \)

Learn how to simplify a complex rational expression of the form \( \frac{1 + \frac{3}{w - 2}}{1 - \frac{3}{w + 2}} \) step-by-step. The solution involves finding common denominators and using reciprocal multiplication. Suitable for high school students.

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