https://file.aiquickdraw.com/mathbot/file/1728875307458kd4egksa.jpg

The expression given in the image is:

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

We can simplify this by expanding the expression using the distributive property (also known as FOIL in algebra).

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

Now, expand each term:

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

First, simplify each part:

\[
= 20 + 15\sqrt{2} - 8\sqrt{2} - 2(3)(\sqrt{2}\sqrt{2})
\]

Since \(\sqrt{2} \times \sqrt{2} = 2\), we get:

\[
= 20 + 15\sqrt{2} - 8\sqrt{2} - 6(2)
\]

Simplify further:

\[
= 20 + 15\sqrt{2} - 8\sqrt{2} - 12
\]

Now, combine like terms:

\[
= (20 - 12) + (15\sqrt{2} - 8\sqrt{2})
\]

\[
= 8 + 7\sqrt{2}
\]

Thus, the simplified form of the expression is:

\[
8 + 7\sqrt{2}
\]

Would you like further details on any part of the simplification? Here are five related questions you might find useful:

1. How do you apply the distributive property to binomials with radicals?
2. Why does \(\sqrt{2} \times \sqrt{2} = 2\)?
3. What are the common mistakes when simplifying expressions with radicals?
4. How do you add or subtract terms involving square roots?
5. What other methods exist for simplifying radical expressions?

**Tip**: Always combine like terms carefully, especially when working with both constants and radicals.

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Learn how to simplify the expression (5 - 2√2)(4 + 3√2) using the distributive property. This detailed solution breaks down each step, covering key algebraic concepts such as radicals, binomial expansion, and the distributive property. Suitable for students in Grades 10-12.

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