Let's first solve for the product of [tex]\((x+3)(x-3)\)[/tex].
Step-by-step Solution:
1. Distribute each term in the first binomial to each term in the second binomial.
[tex]\[
(x + 3)(x - 3)
\][/tex]
2. Apply the distributive property (also known as the FOIL method for binomials):
[tex]\[
(x + 3)(x - 3) = x \cdot x + x \cdot (-3) + 3 \cdot x + 3 \cdot (-3)
\][/tex]
3. Calculate each term:
- [tex]\(x \cdot x = x^2\)[/tex]
- [tex]\(x \cdot (-3) = -3x\)[/tex]
- [tex]\(3 \cdot x = 3x\)[/tex]
- [tex]\(3 \cdot (-3) = -9\)[/tex]
4. Combine all the terms:
[tex]\[
x^2 + (-3x) + (3x) + (-9)
\][/tex]
5. Simplify the combined expression by combining like terms:
- [tex]\(-3x + 3x = 0\)[/tex]
So, you are left with:
[tex]\[
x^2 - 9
\][/tex]
Therefore, the product of [tex]\((x+3)(x-3)\)[/tex] is:
[tex]\[
x^2 - 9
\][/tex]
Now, let's match our result with the given multiple-choice options:
- [tex]\(x^2 - 6x + 9\)[/tex]
- [tex]\(x^2 + 6x + 9\)[/tex]
- [tex]\(x^2 + 9\)[/tex]
- [tex]\(x^2 - 9\)[/tex]
The correct choice is:
[tex]\[
\boxed{x^2 - 9}
\][/tex]
