To simplify the expression \( \frac{3x - 6}{4x^2 - 24x + 32} \), let's go through the steps.

### Step 1: Simplify the numerator
The numerator is \( 3x - 6 \). We can factor this expression:
\[
3x - 6 = 3(x - 2)
\]

### Step 2: Simplify the denominator
The denominator is \( 4x^2 - 24x + 32 \). We can factor out the common factor of 4:
\[
4x^2 - 24x + 32 = 4(x^2 - 6x + 8)
\]
Now, we need to factor the quadratic \( x^2 - 6x + 8 \).

To factor \( x^2 - 6x + 8 \), we look for two numbers that multiply to 8 and add to -6. The numbers -4 and -2 work:
\[
x^2 - 6x + 8 = (x - 4)(x - 2)
\]
Thus, the denominator becomes:
\[
4(x - 4)(x - 2)
\]

### Step 3: Simplify the expression
Now, substitute the factored forms back into the original expression:
\[
\frac{3(x - 2)}{4(x - 4)(x - 2)}
\]
We can cancel the common factor \( x - 2 \) from the numerator and denominator:
\[
\frac{3}{4(x - 4)}
\]

### Final Answer:
The simplified expression is:
\[
\frac{3}{4(x - 4)}
\]

Would you like any further details or have any questions about this solution? 

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Here are 5 questions related to the topic:
1. How would the expression change if the numerator were \( 3x + 6 \)?
2. What if the quadratic in the denominator was \( x^2 - 6x + 9 \)?
3. How would you solve for \( x \) if the expression were set equal to a number?
4. What if the original expression had a negative sign in front of it?
5. How can you verify the factorization of a quadratic expression?

**Tip:** Always double-check the common factors before canceling to ensure that you simplify correctly.

Learn how to simplify the expression (3x - 6) / (4x^2 - 24x + 32) step-by-step with detailed explanations and factorization techniques. This problem involves algebraic manipulation and factoring of quadratic expressions. Suitable for high school students studying algebra.

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