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You've uploaded a set of mathematical expressions to be simplified step by step. I'll break down the first few and explain the simplification process.

### Problem 1:
\[
\left( \frac{1}{a-2} - \frac{4}{a^2 - 4} - \frac{1}{2a + 4} \right) \div \frac{1}{(a+2)^2}
\]

#### Step-by-step simplification:

1. **Factor the quadratic expressions**:
   - \( a^2 - 4 = (a - 2)(a + 2) \)
   - \( 2a + 4 = 2(a + 2) \)
   - \( (a + 2)^2 \) stays as it is.

2. **Rewrite the expression**:
   \[
   \left( \frac{1}{a - 2} - \frac{4}{(a - 2)(a + 2)} - \frac{1}{2(a + 2)} \right) \div \frac{1}{(a + 2)^2}
   \]

3. **Find a common denominator for the terms inside the parentheses**:
   - The common denominator is \( 2(a - 2)(a + 2) \).

   Convert each term:
   \[
   \frac{1}{a - 2} = \frac{2(a + 2)}{2(a - 2)(a + 2)}
   \]
   \[
   \frac{4}{(a - 2)(a + 2)} = \frac{4}{(a - 2)(a + 2)}
   \]
   \[
   \frac{1}{2(a + 2)} = \frac{a - 2}{2(a - 2)(a + 2)}
   \]

4. **Combine the terms**:
   \[
   \frac{2(a + 2) - 4 - (a - 2)}{2(a - 2)(a + 2)}
   \]
   Simplify the numerator:
   \[
   2a + 4 - 4 - a + 2 = a + 6
   \]
   So the combined expression is:
   \[
   \frac{a + 6}{2(a - 2)(a + 2)}
   \]

5. **Divide by \( \frac{1}{(a + 2)^2} \)**:
   This becomes multiplication:
   \[
   \frac{a + 6}{2(a - 2)(a + 2)} \times (a + 2)^2 = \frac{(a + 6)(a + 2)}{2(a - 2)}
   \]

This is the simplified form of the first expression.

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Would you like me to proceed to the next problem or explain any part in more detail? Here's a set of relative questions to explore this further:

1. What are the key strategies to handle fractional expressions with polynomials?
2. How do you factor quadratic expressions effectively?
3. Why is finding a common denominator important when simplifying expressions?
4. How do you manage division in expressions that involve fractions?
5. How does simplification of rational expressions differ in more complex problems?

### Tip: 
When simplifying complex rational expressions, always check for opportunities to factor and cancel terms, especially with quadratics like \( a^2 - b^2 \).

Simplify the following algebraic expressions with polynomials and rational expressions, such as (1/(a-2) - 4/(a^2-4) - 1/(2a+4)) ÷ 1/(a+2)^2, and others.

Step-by-step simplification of rational algebraic expressions involving factoring, common denominators, and polynomials. Learn to simplify complex algebraic expressions like (1/(a-2) - 4/(a^2-4) - 1/(2a+4)) ÷ 1/(a+2)^2, suitable for students in Grades 9-12.

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