https://file.aiquickdraw.com/mathbot/file/1725458542309yal6v1ht.jpg

The problem shown in the image is to simplify the expression:

\[
\frac{4x^2 - 6x}{4x^2 - 9}
\]

Let's break it down step by step:

### Step 1: Factor the numerator and denominator.

- **Numerator**: \(4x^2 - 6x\) can be factored by taking out the common factor of \(2x\):

\[
4x^2 - 6x = 2x(2x - 3)
\]

- **Denominator**: \(4x^2 - 9\) is a difference of squares, which can be factored as:

\[
4x^2 - 9 = (2x - 3)(2x + 3)
\]

### Step 2: Write the expression in its factored form.

\[
\frac{2x(2x - 3)}{(2x - 3)(2x + 3)}
\]

### Step 3: Simplify by canceling common factors.

Since \(2x - 3\) appears in both the numerator and the denominator, we can cancel it out (as long as \(2x - 3 \neq 0\), which means \(x \neq \frac{3}{2}\)):

\[
\frac{2x}{2x + 3}
\]

### Final Simplified Expression:

\[
\frac{2x}{2x + 3}
\]

This is the fully simplified form of the given expression.

Would you like more details, or do you have any questions?

Here are some related questions to consider:

1. How would the expression change if the denominator were \(4x^2 + 9\) instead of \(4x^2 - 9\)?
2. What is the domain of the simplified expression \(\frac{2x}{2x + 3}\)?
3. How would you simplify \(\frac{4x^2 - 6x + 1}{4x^2 - 9}\)?
4. What is the significance of the difference of squares in factoring?
5. Can you identify any values of \(x\) that are not allowed in the original expression?

**Tip**: Always check for any restrictions in the domain of your simplified expression, particularly where the original denominator could be zero.

Simplify the expression: (4x^2 - 6x) / (4x^2 - 9)

Learn how to simplify the rational expression (4x^2 - 6x) / (4x^2 - 9) using algebraic techniques such as factoring quadratics and applying the difference of squares theorem. This step-by-step guide is suitable for students in Grades 9-10.

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