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The image you uploaded contains several algebraic fraction problems focused on division and simplification of fractions. Let's go step by step to simplify each expression.

### 1st Problem:
\[
\frac{m^2 - 4n^2}{2m + n} \div \frac{2mn + 4n^2}{2m^2 + nm}
\]

1. **Step 1**: Factor both numerators and denominators where possible:
   - \(m^2 - 4n^2\) is a difference of squares: \(m^2 - 4n^2 = (m - 2n)(m + 2n)\).
   - \(2mn + 4n^2 = 2n(m + 2n)\).
   - \(2m^2 + nm = m(2m + n)\).

2. **Step 2**: Rewrite the expression using multiplication by the reciprocal:
\[
\frac{(m - 2n)(m + 2n)}{2m + n} \times \frac{m(2m + n)}{2n(m + 2n)}
\]

3. **Step 3**: Cancel out common factors:
   - Cancel \(2m + n\) and \(m + 2n\).

The simplified form is:
\[
\frac{m(m - 2n)}{2n}
\]

---

### 2nd Problem:
\[
\frac{\frac{4a^2}{9a^3 + 3a^2 + a}} + \frac{2 - 6a}{27a^3 - 1}
\]

1. **Step 1**: Factor the denominators:
   - \(9a^3 + 3a^2 + a = a(9a^2 + 3a + 1)\).
   - \(27a^3 - 1\) is a difference of cubes: \(27a^3 - 1 = (3a - 1)(9a^2 + 3a + 1)\).

2. **Step 2**: Rewrite the fractions:
\[
\frac{4a^2}{a(9a^2 + 3a + 1)} + \frac{2 - 6a}{(3a - 1)(9a^2 + 3a + 1)}
\]

3. **Step 3**: Find a common denominator and simplify the numerators. We can combine the fractions over the common denominator \((a)(3a - 1)(9a^2 + 3a + 1)\).

I'll need to continue simplifying this expression, but this would involve combining the numerators and possibly simplifying further.

---

### 3rd Problem:
\[
\frac{(x + 1)(x + 3) + 1}{(x - 1)(x - 3) + 1} \div \frac{(x + 2)^2}{(x - 2)^2}
\]

1. **Step 1**: Simplify both expressions individually:
   - Expand the numerator \((x + 1)(x + 3) + 1 = x^2 + 4x + 3 + 1 = x^2 + 4x + 4\).
   - Expand the denominator \((x - 1)(x - 3) + 1 = x^2 - 4x + 3 + 1 = x^2 - 4x + 4\).

2. **Step 2**: Rewrite the division as multiplication by the reciprocal:
\[
\frac{x^2 + 4x + 4}{x^2 - 4x + 4} \times \frac{(x - 2)^2}{(x + 2)^2}
\]

3. **Step 3**: Simplify:
   - Notice \(x^2 + 4x + 4 = (x + 2)^2\) and \(x^2 - 4x + 4 = (x - 2)^2\).
   - The expression simplifies to:
\[
\frac{(x + 2)^2}{(x - 2)^2} \times \frac{(x - 2)^2}{(x + 2)^2}
\]

All terms cancel, leaving \(1\) as the result.

---

Would you like me to continue with the other problems in the image? Feel free to ask for further clarification or details on any of the steps above!

Here are 5 related questions:
1. How do you factor a difference of squares?
2. What is the process of simplifying complex fractions?
3. Can all algebraic fractions be simplified? Why or why not?
4. How do you find a common denominator for adding fractions?
5. What are the key steps in dividing algebraic fractions?

**Tip:** Always look for common factors in both numerators and denominators before performing any operations when simplifying algebraic fractions.

The original question submitted by the user as text is based on algebraic division and simplification of fractions from Exercise 3.4.

Explore how to simplify algebraic fractions through division and factorization. This exercise from Section 3.4 includes steps such as factoring difference of squares and canceling common terms. Suitable for students in Grades 9-12.

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