https://file.aiquickdraw.com/mathbot/file/17253804478926bcbsk8y.jpg

The image contains a mathematical equation, and I will extract and solve it.

The equation appears to be:

\[
\frac{6a^2 + 5a - 6}{4a^2 - 9} \times \frac{2a + 3}{3a - 2} \div \left(\frac{4a - 6}{3a + 4}\right) = ?
\]

Let's break this down and solve it step by step.

### Step 1: Simplify each fraction where possible
1. **Factor the numerators and denominators**:
   - \(6a^2 + 5a - 6\) can be factored into \((2a - 1)(3a + 6)\).
   - \(4a^2 - 9\) is a difference of squares and factors to \((2a - 3)(2a + 3)\).
   - \(4a - 6\) can be factored to \(2(2a - 3)\).

The equation now becomes:

\[
\frac{(2a - 1)(3a + 6)}{(2a - 3)(2a + 3)} \times \frac{2a + 3}{3a - 2} \div \frac{2(2a - 3)}{3a + 4}
\]

### Step 2: Multiply and divide the fractions
- When dividing by a fraction, we multiply by its reciprocal. Thus, the division in the equation becomes:

\[
\frac{(2a - 1)(3a + 6)}{(2a - 3)(2a + 3)} \times \frac{2a + 3}{3a - 2} \times \frac{3a + 4}{2(2a - 3)}
\]

### Step 3: Simplify the expression
- **Cancel out common factors**:
   - The term \(2a + 3\) appears in the numerator and denominator, so it cancels out.
   - The term \(2a - 3\) also appears in both, so it cancels out.

The equation simplifies to:

\[
\frac{2a - 1}{3a - 2} \times \frac{3a + 4}{2}
\]

### Step 4: Multiply the remaining terms

Now, we multiply the remaining fractions:

\[
\frac{(2a - 1)(3a + 4)}{2(3a - 2)}
\]

Expanding both the numerator and denominator:

- **Numerator**: \((2a - 1)(3a + 4)\) expands to \(6a^2 + 8a - 3a - 4 = 6a^2 + 5a - 4\).
- **Denominator**: \(2(3a - 2)\) expands to \(6a - 4\).

So the final simplified form is:

\[
\frac{6a^2 + 5a - 4}{6a - 4}
\]

This is the simplified expression. 

Would you like further details, or do you have any questions about this process? Here are some related questions that might interest you:

1. What is the difference between multiplying and dividing fractions in algebra?
2. How do you factor quadratic expressions efficiently?
3. What is the significance of the difference of squares in simplifying expressions?
4. How do you handle more complex rational expressions in algebra?
5. How do you identify and cancel common factors in rational expressions?

**Tip:** Always factor both the numerator and denominator completely to simplify rational expressions effectively.

Simplify the expression: \[ \frac{6a^2 + 5a - 6}{4a^2 - 9} \times \frac{2a + 3}{3a - 2} \div \left(\frac{4a - 6}{3a + 4}\right) \]

Learn how to simplify complex rational expressions using factoring and division. This guide provides a detailed step-by-step solution to simplifying expressions involving multiplication and division of fractions. Suitable for students in Grades 9-12.

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