Let's break down and simplify the given expression step by step.

Expression:  
\[
\sqrt{20} + 44.5 + \sqrt{45} - 4\sqrt{20}
\]

### Step 1: Simplify the square roots
- \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\)
- \(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\)
- \(4\sqrt{20} = 4 \times 2\sqrt{5} = 8\sqrt{5}\)

### Step 2: Substitute the simplified square roots back into the expression:
\[
2\sqrt{5} + 44.5 + 3\sqrt{5} - 8\sqrt{5}
\]

### Step 3: Combine like terms (the terms with \(\sqrt{5}\)):
\[
(2\sqrt{5} + 3\sqrt{5} - 8\sqrt{5}) + 44.5
\]
\[
(5\sqrt{5} - 8\sqrt{5}) + 44.5
\]
\[
-3\sqrt{5} + 44.5
\]

So, the simplified expression is:
\[
44.5 - 3\sqrt{5}
\]

This is the final simplified form of the expression.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you simplify square roots involving non-perfect squares?
2. What are the steps for combining like terms with radicals?
3. How do you rationalize a denominator with a square root?
4. Can you add or subtract different square roots directly?
5. How would the result change if one of the terms was a cube root instead of a square root?

**Tip:** When simplifying square roots, always check if the number can be factored into a perfect square times another number to simplify easily.

Explore how to simplify the expression √20 + 44.5 + √45 − 4√20 step by step. This guide explains simplifying square roots and combining like terms, suitable for students in Grades 9-11.

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