To find the product [tex]\((3 - 2i)(3 + 2i)\)[/tex], we will use the distributive property of multiplication over addition. Follow these detailed steps:
1. Start by expanding the expression using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(3 - 2i)(3 + 2i) = 3(3) + 3(2i) - 2i(3) - 2i(2i)
\][/tex]
2. Multiply the terms:
[tex]\[
= 3 \cdot 3 + 3 \cdot 2i - 2i \cdot 3 - 2i \cdot 2i
\][/tex]
[tex]\[
= 9 + 6i - 6i - 4i^2
\][/tex]
3. Notice that [tex]\(6i\)[/tex] and [tex]\(-6i\)[/tex] cancel each other out:
[tex]\[
= 9 - 4i^2
\][/tex]
4. Recall that [tex]\(i^2 = -1\)[/tex]. Therefore, [tex]\(-4i^2\)[/tex] becomes [tex]\(-4(-1)\)[/tex]:
[tex]\[
= 9 + 4
\][/tex]
5. Finally, add the real parts together:
[tex]\[
= 13
\][/tex]
Thus, the value of the product [tex]\((3 - 2i)(3 + 2i)\)[/tex] is [tex]\(13\)[/tex].
So the correct answer is:
[tex]\[ \boxed{13} \][/tex]
