## Factoring the Expression (x^2 - 4)(x^2 + 6x + 9)

This expression represents the product of two binomials. To factor it completely, we need to factor each binomial individually.

### Factoring (x^2 - 4)

This binomial is a **difference of squares**, where:

- x^2 is the square of x
- 4 is the square of 2

The difference of squares pattern is: **a^2 - b^2 = (a + b)(a - b)**

Applying this pattern to our binomial:

- a = x
- b = 2

Therefore, **(x^2 - 4) = (x + 2)(x - 2)**

### Factoring (x^2 + 6x + 9)

This binomial is a **perfect square trinomial**, where:

- The first term (x^2) is the square of x
- The last term (9) is the square of 3
- The middle term (6x) is twice the product of x and 3 (2 * x * 3 = 6x)

The perfect square trinomial pattern is: **a^2 + 2ab + b^2 = (a + b)^2**

Applying this pattern to our binomial:

- a = x
- b = 3

Therefore, **(x^2 + 6x + 9) = (x + 3)^2**

### Combining the Factors

Now that we have factored both binomials, we can substitute them back into the original expression:

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

**This is the completely factored form of the expression.**