1. **State the problem:** Simplify the expression $2\sqrt{45} + 2\sqrt{90} + 3\sqrt{45}$. 2. **Recall the rule:** $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and simplify square roots by factoring out perfect squares. 3. **Simplify each square root:** - $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$ - $\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}$ 4. **Substitute back:** $$2\sqrt{45} + 2\sqrt{90} + 3\sqrt{45} = 2(3\sqrt{5}) + 2(3\sqrt{10}) + 3(3\sqrt{5})$$ 5. **Multiply coefficients:** $$= 6\sqrt{5} + 6\sqrt{10} + 9\sqrt{5}$$ 6. **Combine like terms:** $$6\sqrt{5} + 9\sqrt{5} = (6 + 9)\sqrt{5} = 15\sqrt{5}$$ 7. **Final simplified expression:** $$15\sqrt{5} + 6\sqrt{10}$$ This is the simplest form since $\sqrt{5}$ and $\sqrt{10}$ are unlike terms and cannot be combined further.