Question: Write the following expression as a single complex number.
$$(3 - 2i)(4 + 2i)$$
Options:
$16 - 2i$
$16 + 2i$
$12 - 4i$
$12 + 4i$
1. **State the problem:** We need to multiply the complex numbers $3 - 2i$ and $4 + 2i$ and write the result as a single complex number.
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:**
$$ (3 - 2i)(4 + 2i) = 3 \times 4 + 3 \times 2i - 2i \times 4 - 2i \times 2i $$
$$ = 12 + 6i - 8i - 4i^2 $$
4. **Simplify the imaginary terms:**
$$ 6i - 8i = -2i $$
5. **Simplify the $i^2$ term:**
Since $i^2 = -1$,
$$ -4i^2 = -4 \times (-1) = 4 $$
6. **Combine all terms:**
$$ 12 - 2i + 4 = (12 + 4) - 2i = 16 - 2i $$
7. **Final answer:**
The product is $$16 - 2i$$.
Therefore, the correct choice is **$16 - 2i$**.