Question: 3 — Multiply Complex Numbers
LEARNING OBJECTIVE: Multiply two complex numbers.
Select the product $$(4 + i)(-2 - i)$$ in the standard form $a + bi$.
a.) $8 - i^2$
b.) $-7 - 6i$
c.) $-8 - i^2$
d.) $7 - 6i$
1. **State the problem:** Multiply the complex numbers $$(4 + i)$$ and $$(-2 - i)$$ and express the result in the standard form $$a + bi$$.
2. **Recall the formula:** To multiply two complex numbers $$(a + bi)(c + di)$$, use the distributive property:
$$
(a + bi)(c + di) = ac + adi + bci + bdi^2
$$
Remember that $$i^2 = -1$$.
3. **Apply the formula:**
$$
(4 + i)(-2 - i) = 4 \times (-2) + 4 \times (-i) + i \times (-2) + i \times (-i)
$$
$$
= -8 - 4i - 2i - i^2
$$
4. **Combine like terms:**
$$
-8 - (4i + 2i) - i^2 = -8 - 6i - i^2
$$
5. **Substitute $$i^2 = -1$$:**
$$
-8 - 6i - (-1) = -8 - 6i + 1
$$
6. **Simplify:**
$$
-8 + 1 - 6i = -7 - 6i
$$
**Final answer:** The product is $$\boxed{-7 - 6i}$$, which corresponds to option b.)