Let's simplify the given expression:

\[
\frac{4a^2 - 32a + 64}{6a^2 - 96}
\]

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

#### Numerator: \( 4a^2 - 32a + 64 \)

This is a quadratic expression, and we can factor out the common factor of 4:

\[
4(a^2 - 8a + 16)
\]

Now, factor \( a^2 - 8a + 16 \). This is a perfect square trinomial:

\[
a^2 - 8a + 16 = (a - 4)^2
\]

So, the numerator becomes:

\[
4(a - 4)^2
\]

#### Denominator: \( 6a^2 - 96 \)

Factor out the common factor of 6:

\[
6(a^2 - 16)
\]

Now, factor \( a^2 - 16 \) as a difference of squares:

\[
a^2 - 16 = (a - 4)(a + 4)
\]

So, the denominator becomes:

\[
6(a - 4)(a + 4)
\]

### Step 2: Simplify the fraction.

Now, rewrite the expression with the factored forms:

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

Cancel out the common factor \( (a - 4) \) from both the numerator and denominator (note that \( a \neq 4 \), to avoid division by zero):

\[
\frac{4(a - 4)}{6(a + 4)}
\]

### Step 3: Simplify the remaining terms.

Now simplify the coefficient:

\[
\frac{4}{6} = \frac{2}{3}
\]

So, the simplified expression is:

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

This is the simplified form of the given expression.

---

Would you like further details or have any questions?  
Here are five related questions to explore:

1. How can we check if the factored form is equivalent to the original?
2. What happens if \( a = 4 \) in the original expression?
3. How does this simplification change if \( a^2 - 16 \) were not a difference of squares?
4. Can you factor and simplify a more complex expression involving cubic terms?
5. What would the graph of this rational expression look like?

**Tip:** Always check for factors common to both the numerator and denominator when simplifying rational expressions.

This lesson explains how to simplify the rational expression (4a^2 - 32a + 64) / (6a^2 - 96) by factoring both the numerator and denominator. Learn key algebraic concepts such as perfect square trinomials, difference of squares, and simplification techniques. Suitable for students in Grades 9-11.

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