Let's simplify the given expression, $-4\sqrt{45} - 10\sqrt{80}$.

Step 1: Simplify the square roots

• $\sqrt{45}$ can be simplified: $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$

• $\sqrt{80}$ can be simplified: $\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}$

Step 2: Substitute back into the expression

Substituting these simplified forms into the original expression: $-4\sqrt{45} - 10\sqrt{80} = -4(3\sqrt{5}) - 10(4\sqrt{5}) = -12\sqrt{5} - 40\sqrt{5}$

Step 3: Combine like terms

Both terms have $\sqrt{5}$ as a common factor, so we can combine them: $-12\sqrt{5} - 40\sqrt{5} = (-12 - 40)\sqrt{5} = -52\sqrt{5}$

Thus, the simplified form of the expression is: $\boxed{-52\sqrt{5}}$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you determine which factors to simplify under a square root?
2. Can you combine terms with different square root factors, like $\sqrt{5}$ and $\sqrt{3}$?
3. What is the difference between simplifying square roots and rationalizing the denominator?
4. How would the result change if there were addition instead of subtraction between the terms?
5. How can you simplify square roots that involve variables, like $\sqrt{a^2b}$?

Tip: When simplifying radicals, always factor out perfect squares first to make simplification easier.