To multiply the complex numbers [tex]\( (3 - 4i) \)[/tex] and [tex]\( (-2 - 2i) \)[/tex], follow these steps:
1. Express the multiplication in distributed form:
[tex]\[
(3 - 4i)(-2 - 2i) = 3(-2 - 2i) + (-4i)(-2 - 2i)
\][/tex]
2. Distribute each term in the parentheses:
- For [tex]\(3(-2 - 2i)\)[/tex]:
[tex]\[
3(-2) + 3(-2i) = -6 - 6i
\][/tex]
- For [tex]\((-4i)(-2 - 2i)\)[/tex]:
[tex]\[
(-4i)(-2) + (-4i)(-2i) = 8i + 8i^2
\][/tex]
3. Recall that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
8i^2 = 8(-1) = -8
\][/tex]
4. Combine the results:
[tex]\[
-6 - 6i + 8i - 8
\][/tex]
5. Combine the real parts and the imaginary parts:
- Combine the real parts:
[tex]\[
-6 - 8 = -14
\][/tex]
- Combine the imaginary parts:
[tex]\[
-6i + 8i = 2i
\][/tex]
Therefore, the product of the complex numbers [tex]\( (3 - 4i) \)[/tex] and [tex]\( (-2 - 2i) \)[/tex] is:
[tex]\[
\boxed{(-14 + 2i)}
\][/tex]
