Question: Multiply. $2i \left(-4 - i\right)^2$
1. **State the problem:** Multiply and simplify the expression $2i \left(-4 - i\right)^2$.
2. **Recall the formula:** To square a binomial, use the formula $\left(a + b\right)^2 = a^2 + 2ab + b^2$.
3. **Apply the formula:** Here, $a = -4$ and $b = -i$.
$$\left(-4 - i\right)^2 = (-4)^2 + 2(-4)(-i) + (-i)^2$$
4. **Calculate each term:**
$$(-4)^2 = 16$$
$$2(-4)(-i) = 8i$$
$$(-i)^2 = (-1)^2 \cdot i^2 = 1 \cdot (-1) = -1$$
5. **Sum the terms:**
$$16 + 8i - 1 = 15 + 8i$$
6. **Multiply by $2i$:**
$$2i \times (15 + 8i) = 2i \times 15 + 2i \times 8i = 30i + 16i^2$$
7. **Recall that $i^2 = -1$:**
$$16i^2 = 16 \times (-1) = -16$$
8. **Combine terms:**
$$30i - 16$$
9. **Write in standard form $a + bi$:**
$$-16 + 30i$$
**Final answer:** $-16 + 30i$