n^2-9            n^2-7n+12      n+4
--------   .  -----------  .  ------
n2-6n+9       n^2+8n+16      n+3

The given expression is:

\[
\frac{n^2 - 9}{n^2 - 6n + 9} \cdot \frac{n^2 - 7n + 12}{n^2 + 8n + 16} \cdot \frac{n + 4}{n + 3}
\]

Let's simplify this step by step.

### Step 1: Factor the numerators and denominators
1. **Factor \(n^2 - 9\):**
   \[
   n^2 - 9 = (n - 3)(n + 3) \quad \text{(difference of squares)}
   \]

2. **Factor \(n^2 - 6n + 9\):**
   \[
   n^2 - 6n + 9 = (n - 3)(n - 3) \quad \text{(perfect square)}
   \]

3. **Factor \(n^2 - 7n + 12\):**
   \[
   n^2 - 7n + 12 = (n - 3)(n - 4)
   \]

4. **Factor \(n^2 + 8n + 16\):**
   \[
   n^2 + 8n + 16 = (n + 4)(n + 4) \quad \text{(perfect square)}
   \]

Now the expression looks like:

\[
\frac{(n - 3)(n + 3)}{(n - 3)(n - 3)} \cdot \frac{(n - 3)(n - 4)}{(n + 4)(n + 4)} \cdot \frac{n + 4}{n + 3}
\]

### Step 2: Cancel common terms
1. In the first fraction, \((n - 3)\) appears in both the numerator and denominator, so cancel one of them:
   \[
   \frac{(n + 3)}{(n - 3)} \cdot \frac{(n - 3)(n - 4)}{(n + 4)(n + 4)} \cdot \frac{n + 4}{n + 3}
   \]

2. Now, cancel \((n + 3)\) from the first and third fractions:
   \[
   \frac{1}{(n - 3)} \cdot \frac{(n - 3)(n - 4)}{(n + 4)(n + 4)} \cdot 1
   \]

3. Cancel one \((n - 3)\) term:
   \[
   \frac{n - 4}{(n + 4)(n + 4)}
   \]

### Final Simplified Expression:
\[
\frac{n - 4}{(n + 4)^2}
\]

This is the fully simplified form of the given expression.

Would you like any further explanation?

Here are five related questions:
1. How do you factor quadratic expressions effectively?
2. Why are perfect square trinomials important in factoring?
3. How do you recognize the difference of squares?
4. What is the significance of canceling common terms in rational expressions?
5. How would this process change if \(n\) were a specific number?

**Tip**: Always look for common factors in both the numerator and the denominator to simplify expressions quickly!

Learn how to simplify the rational expression involving n^2 - 9, n^2 - 7n + 12, and other quadratic terms. This problem involves factoring quadratic expressions, identifying perfect squares, and simplifying fractions. Suitable for students in Grades 9-10.

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