https://r2.erweima.ai/mathbot/image/17259515634550xgkipay.jpg

The expression given in the image is:

\[
(2\sqrt{3} - 5)(\sqrt{2} + 4)
\]

Now, let's simplify this expression using the distributive property (FOIL method).

### Step 1: Multiply each term of the first expression by each term of the second expression.

\[
(2\sqrt{3})(\sqrt{2}) + (2\sqrt{3})(4) + (-5)(\sqrt{2}) + (-5)(4)
\]

### Step 2: Perform the multiplication.

- \( (2\sqrt{3})(\sqrt{2}) = 2\sqrt{6} \)
- \( (2\sqrt{3})(4) = 8\sqrt{3} \)
- \( (-5)(\sqrt{2}) = -5\sqrt{2} \)
- \( (-5)(4) = -20 \)

### Step 3: Combine the terms.

\[
2\sqrt{6} + 8\sqrt{3} - 5\sqrt{2} - 20
\]

This is the simplified form of the expression:
\[
2\sqrt{6} + 8\sqrt{3} - 5\sqrt{2} - 20
\]

Would you like further details on this solution or have any questions?

Here are some related questions for further exploration:
1. How would the result change if we swapped the terms in the parentheses?
2. Can we combine any of the square root terms further?
3. How would you apply the distributive property in a more complex expression?
4. What are the properties of radicals that allow us to simplify square root terms?
5. What is the effect of multiplying constants with square roots?

**Tip**: When multiplying expressions with square roots, remember to treat the square root terms as variables during the expansion and then simplify at the end.

Learn how to simplify the expression (2√3 - 5)(√2 + 4) step by step using the distributive property. This problem covers key algebraic concepts such as multiplication of square roots and radical simplification. Suitable for students in Grades 8-10.

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