To simplify the given expression:

$\frac{1 + \frac{y}{x+2}}{1 - \frac{y^2}{x+2}}$

Step 1: Rewrite the numerator and denominator

The numerator is: $1 + \frac{y}{x+2} = \frac{x+2}{x+2} + \frac{y}{x+2} = \frac{x+2 + y}{x+2}.$

The denominator is: $1 - \frac{y^2}{x+2} = \frac{x+2}{x+2} - \frac{y^2}{x+2} = \frac{x+2 - y^2}{x+2}.$

Step 2: Combine the terms into a single fraction

The entire expression becomes: $\frac{\frac{x+2 + y}{x+2}}{\frac{x+2 - y^2}{x+2}}.$

Step 3: Simplify the division

Dividing two fractions is equivalent to multiplying by the reciprocal: $\frac{x+2 + y}{x+2} \times \frac{x+2}{x+2 - y^2}.$

The $x+2$ in the numerator and denominator cancels out, leaving: $\frac{x+2 + y}{x+2 - y^2}.$

Final Answer:

$\frac{x+2 + y}{x+2 - y^2}.$

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Would you like me to explain further or clarify any step?

Here are five related questions for practice:

1. How would you simplify $\frac{1 + \frac{a}{b}}{1 - \frac{c}{b}}$?
2. Can you find the domain of the simplified expression?
3. What happens if $x+2 - y^2 = 0$ in the denominator?
4. How does the simplification change if $y^2 = 1$?
5. Could this expression represent a real-world problem? If so, provide an example.

Tip: Always check for restrictions in the denominator when simplifying algebraic expressions!