## Factoring and Simplifying the Expression (x² - 4)(x² + 6x + 9)

This expression represents the product of two quadratic expressions. To simplify it, we can factor each quadratic expression and then multiply the resulting factors.

### Factoring the First Expression: (x² - 4)

This expression is a **difference of squares**, which factors as follows:

**(a² - b²) = (a + b)(a - b)**

Applying this to (x² - 4), we get:

**(x² - 4) = (x + 2)(x - 2)**

### Factoring the Second Expression: (x² + 6x + 9)

This expression is a **perfect square trinomial**, which factors as follows:

**(a² + 2ab + b²) = (a + b)²**

Applying this to (x² + 6x + 9), we get:

**(x² + 6x + 9) = (x + 3)²**

### Multiplying the Factored Expressions

Now we have:

**(x² - 4)(x² + 6x + 9) = (x + 2)(x - 2)(x + 3)²**

This is the fully factored form of the original expression.

### Expanding the Expression (Optional)

While the factored form is often the most useful, we can also expand the expression to obtain a polynomial in standard form:

**(x + 2)(x - 2)(x + 3)² = (x² - 4)(x² + 6x + 9)**

Expanding the product, we get:

**(x² - 4)(x² + 6x + 9) = x⁴ + 6x³ + 5x² - 24x - 36**

This is the simplified polynomial form of the expression.

### Conclusion

The expression (x² - 4)(x² + 6x + 9) can be simplified by factoring each quadratic expression. This gives us the factored form: (x + 2)(x - 2)(x + 3)², which can be further expanded to obtain the polynomial form: x⁴ + 6x³ + 5x² - 24x - 36.