Question: Simplify the following
radical expression.
$(5 + 2\sqrt{2})(3\sqrt{2} - 3)$
$-3 + [?]\sqrt{[ ]}$
position_hint: center
1. **State the problem:** Simplify the expression $$(5 + 2\sqrt{2})(3\sqrt{2} - 3)$$.
2. **Use the distributive property (FOIL) to expand:**
$$
(5)(3\sqrt{2}) + (5)(-3) + (2\sqrt{2})(3\sqrt{2}) + (2\sqrt{2})(-3)
$$
3. **Calculate each term:**
- $5 \times 3\sqrt{2} = 15\sqrt{2}$
- $5 \times (-3) = -15$
- $2\sqrt{2} \times 3\sqrt{2} = 2 \times 3 \times \sqrt{2} \times \sqrt{2} = 6 \times 2 = 12$
- $2\sqrt{2} \times (-3) = -6\sqrt{2}$
4. **Combine all terms:**
$$
15\sqrt{2} - 15 + 12 - 6\sqrt{2}
$$
5. **Group like terms:**
$$
(15\sqrt{2} - 6\sqrt{2}) + (-15 + 12) = 9\sqrt{2} - 3
$$
6. **Rewrite the expression:**
$$
-3 + 9\sqrt{2}
$$
**Final answer:** $-3 + 9\sqrt{2}$