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.