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.