Question: Simplify the expression to a + bi form:
$(-4 - 5i)(9 + 2i)$
1. **State the problem:** Simplify the expression $(-4 - 5i)(9 + 2i)$ into the form $a + bi$ where $a$ and $b$ are real numbers.
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:**
$$ (-4 - 5i)(9 + 2i) = (-4)(9) + (-4)(2i) + (-5i)(9) + (-5i)(2i) $$
4. **Calculate each term:**
$$ = -36 - 8i - 45i - 10i^2 $$
5. **Simplify $i^2$ term:**
Since $i^2 = -1$,
$$ -10i^2 = -10(-1) = 10 $$
6. **Combine like terms:**
$$ -36 + 10 - 8i - 45i = (-36 + 10) + (-8i - 45i) = -26 - 53i $$
7. **Final answer:**
The expression simplifies to
$$ \boxed{-26 - 53i} $$