https://r2.erweima.ai/mathbot/image/1726593858034lfizk8ea.jpg

The expression you've uploaded is:

\[
\frac{\frac{x+2}{x-2}}{\frac{x^2 - 4}{x^2 + 8x + 16}}
\]

Let's simplify this step by step.

### Step 1: Recognize factorizations
- The numerator of the second fraction \((x^2 - 4)\) is a difference of squares:
  \[
  x^2 - 4 = (x - 2)(x + 2)
  \]
- The denominator of the second fraction \((x^2 + 8x + 16)\) is a perfect square trinomial:
  \[
  x^2 + 8x + 16 = (x + 4)^2
  \]

### Step 2: Rewrite the expression
The given expression can now be written as:

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

### Step 3: Simplify the division
Dividing by a fraction is the same as multiplying by its reciprocal, so we multiply the first fraction by the reciprocal of the second fraction:

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

### Step 4: Cancel common terms
Now, cancel the common terms:
- The \((x+2)\) in the numerator and denominator cancels out.
- The \((x-2)\) in the numerator and denominator also cancels out.

We are left with:

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

### Final Simplified Expression:
The simplified result is:

\[
(x + 4)^2
\]

Let me know if you'd like further clarification!

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**Related questions:**
1. How do you factorize a difference of squares?
2. What are the steps to simplify a complex rational expression?
3. How does multiplying by the reciprocal work when simplifying fractions?
4. Why is it important to cancel common factors in algebraic expressions?
5. What is the significance of perfect square trinomials in factoring?

**Tip:** Always check for factorizations like difference of squares or perfect square trinomials before simplifying algebraic expressions.

Simplify the expression (x + 2)/(x - 2) ÷ (x^2 - 4)/(x^2 + 8x + 16).

Step-by-step guide on how to simplify the rational expression (x + 2)/(x - 2) ÷ (x^2 - 4)/(x^2 + 8x + 16). Learn how to factor the difference of squares and perfect square trinomials to simplify complex rational expressions. Suitable for students in Grades 9-11.

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