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Let's solve the problems one by one:

### 5. Simplifying the quotient

The expression is:
\[
\frac{a^3 - a^2}{2a^2 + 3a} \div \frac{3a}{2a + 3}, \quad a \neq \frac{3}{2}, 0
\]

1. Simplify the first fraction:
\[
\frac{a^3 - a^2}{2a^2 + 3a} = \frac{a^2(a - 1)}{a(2a + 3)} = \frac{a(a - 1)}{2a + 3}
\]

2. Simplify the division:
\[
\frac{\frac{a(a - 1)}{2a + 3}}{\frac{3a}{2a + 3}} = \frac{a(a - 1)}{2a + 3} \times \frac{2a + 3}{3a} = \frac{a(a - 1)}{3a}
\]

Thus, the final expression is:
\[
\frac{a - 1}{3}
\]

The correct answer is **B**.

---

### 6. Simplifying the sum

The expression is:
\[
\frac{9}{5x^2} + \frac{x - 3}{5x^2}, \quad x \neq 0
\]

Both fractions have the same denominator, so combine the numerators:
\[
\frac{9 + (x - 3)}{5x^2} = \frac{x + 6}{5x^2}
\]

The correct answer is **D**.

---

### 7. Simplifying the difference

The expression is:
\[
\frac{3}{x - 7} - \frac{4}{x^2 - 49}, \quad x \neq \pm 7
\]

Factor the denominator in the second term:
\[
x^2 - 49 = (x - 7)(x + 7)
\]

Rewrite the expression:
\[
\frac{3}{x - 7} - \frac{4}{(x - 7)(x + 7)}
\]

Find a common denominator:
\[
= \frac{3(x + 7) - 4}{(x - 7)(x + 7)} = \frac{3x + 21 - 4}{(x - 7)(x + 7)} = \frac{3x + 17}{(x - 7)(x + 7)}
\]

So, the expression is:
\[
\frac{3x + 17}{(x - 7)(x + 7)}
\]

The values of \(a\) and \(b\) are 3 and 17, respectively.

The correct answer is **D**.

---

Do you want more details on any of these solutions?

Here are 5 related questions:
1. How do you find the least common denominator in rational expressions?
2. What is the process for factoring quadratic expressions?
3. How do you simplify complex fractions?
4. What are the restrictions on the variable in rational expressions?
5. How do you solve rational equations?

**Tip:** When simplifying algebraic fractions, always look for common factors in the numerator and denominator.

Simplify the given algebraic expressions and find the correct answers for the three problems.

Explore how to simplify algebraic expressions involving quotients, sums, and differences. This problem set covers key algebraic concepts like factoring, dividing rational expressions, and combining like terms. Suitable for students in Grades 9-11.

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