Simplify the expression $(x+3)^2 (x-3)^2$.

1. Problem Description

Simplify the expression $(x+3)^2 (x-3)^2$.

2. Solution Steps

First, we can rewrite the expression as follows:

$(x+3)^2 (x-3)^2 = [(x+3)(x-3)]^2$

Now, we can use the difference of squares formula:

$(a+b)(a-b) = a^2 - b^2$

In our case, $a = x$ and $b = 3$, so we have:

$(x+3)(x-3) = x^2 - 3^2 = x^2 - 9$

Substituting this back into the original expression, we get:

$[(x+3)(x-3)]^2 = (x^2 - 9)^2$

Now we need to expand $(x^2 - 9)^2$. We can use the formula $(a-b)^2 = a^2 - 2ab + b^2$, where $a = x^2$ and $b = 9$.

$(x^2 - 9)^2 = (x^2)^2 - 2(x^2)(9) + 9^2$

$(x^2 - 9)^2 = x^4 - 18x^2 + 81$

3. Final Answer

$x^4 - 18x^2 + 81$