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Let's simplify the given expression step by step.

The expression is:

\[
\frac{(x^2 - 16)(x^2 + 9)}{(x - 3)(x + 4)}
\]

### Step 1: Factor the numerator

1. **Factor \(x^2 - 16\):**  
   \(x^2 - 16\) is a difference of squares, which can be factored as:
   \[
   x^2 - 16 = (x - 4)(x + 4)
   \]

2. **Factor \(x^2 + 9\):**  
   \(x^2 + 9\) is a sum of squares, which cannot be factored further over the real numbers. Thus, it remains as \(x^2 + 9\).

So, the numerator becomes:
\[
(x - 4)(x + 4)(x^2 + 9)
\]

### Step 2: Rewrite the entire expression

The expression now becomes:
\[
\frac{(x - 4)(x + 4)(x^2 + 9)}{(x - 3)(x + 4)}
\]

### Step 3: Cancel common factors

Notice that \( (x + 4) \) appears in both the numerator and the denominator. We can cancel it out:
\[
\frac{(x - 4)(x^2 + 9)}{x - 3}
\]

### Final Answer

The simplified expression is:

\[
\frac{(x - 4)(x^2 + 9)}{x - 3}
\]

Would you like more details or have any questions? Here are five related questions you might find useful:

1. How do you factor polynomials involving differences and sums of squares?
2. What should you check before canceling terms in a fraction?
3. How can the sum of squares \(x^2 + 9\) be interpreted in the complex plane?
4. What other types of expressions can be simplified similarly to this one?
5. How would the process differ if there were additional terms in the denominator?

**Tip:** Always check for common factors in both the numerator and denominator before canceling to simplify the expression efficiently.

Learn how to simplify the rational expression ((x^2 - 16)(x^2 + 9)) / ((x - 3)(x + 4)) step-by-step. This problem involves factoring polynomials and cancelling common factors. Suitable for high school students in Grades 10-12 studying algebra and rational expressions.

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