Question: Multiply.
$(5 + 7i)(8 - 2i) = [blank] + [blank] i$
(Simplify your answer. Type integers. Do not factor.)
1. **State the problem:** Multiply the complex numbers $5 + 7i$ and $8 - 2i$.
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:**
$$ (5 + 7i)(8 - 2i) = 5 \times 8 + 5 \times (-2i) + 7i \times 8 + 7i \times (-2i) $$
4. **Calculate each term:**
$$ = 40 - 10i + 56i - 14i^2 $$
5. **Simplify $i^2$ term:**
Since $i^2 = -1$,
$$ -14i^2 = -14 \times (-1) = 14 $$
6. **Combine like terms:**
$$ 40 + 14 + (-10i + 56i) = 54 + 46i $$
7. **Final answer:**
$$ (5 + 7i)(8 - 2i) = 54 + 46i $$