To simplify the expression:

$-10\sqrt{45} + 5\sqrt{20}$

let’s start by simplifying each square root term.

1. Simplify $\sqrt{45}$: $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$ So, $-10\sqrt{45} = -10 \times 3\sqrt{5} = -30\sqrt{5}$

2. Simplify $\sqrt{20}$: $\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$ So, $5\sqrt{20} = 5 \times 2\sqrt{5} = 10\sqrt{5}$

3. Combine the terms: $-30\sqrt{5} + 10\sqrt{5} = (-30 + 10)\sqrt{5} = -20\sqrt{5}$

Final Answer:

$-10\sqrt{45} + 5\sqrt{20} = -20\sqrt{5}$

Would you like more details or have any questions?

Further Questions:

1. How do you simplify square roots in general?
2. Why can square roots of perfect squares be separated out?
3. How does distributing constants work in square root simplification?
4. Are there cases where square roots cannot be simplified further?
5. How can we use this method for complex numbers?

Tip: When simplifying expressions with square roots, look for factors that are perfect squares to simplify square roots easily.