## Factoring the Expression (a+3)(a-3)(a^2+9)

This expression is a classic example of a product of **difference of squares** and **sum of squares**. Let's break down the factorization step-by-step:

### Difference of Squares Pattern

- The first two factors, (a+3) and (a-3), represent the
**difference of squares**. This is because they follow the form (x + y)(x - y) where x = a and y = 3. - Using the difference of squares formula:
**(x + y)(x - y) = x^2 - y^2** - We can simplify the first two factors as follows:
- (a + 3)(a - 3) = a^2 - 3^2 = a^2 - 9

### Sum of Squares Pattern

- The third factor, (a^2 + 9), represents the
**sum of squares**. **Importantly, there is no direct formula for factoring the sum of squares over real numbers.**

### Final Factorization

- Combining the simplified first two factors with the third factor, we get:
**(a^2 - 9)(a^2 + 9)**

- This is the fully factored form of the expression.

### Conclusion

Therefore, the fully factored form of the expression (a+3)(a-3)(a^2+9) is **(a^2 - 9)(a^2 + 9)**. It's important to note that the sum of squares term (a^2 + 9) cannot be factored further over the real numbers.