The given expression to simplify is:

$(\sqrt{2} + 3)(\sqrt{2} - 5)$

This is a product of two binomials, so we use the distributive property (FOIL method) to expand it:

Step 1: Expand

$(\sqrt{2} + 3)(\sqrt{2} - 5) = (\sqrt{2})(\sqrt{2}) + (\sqrt{2})(-5) + (3)(\sqrt{2}) + (3)(-5)$

Step 2: Simplify each term

1. $(\sqrt{2})(\sqrt{2}) = 2$
2. $(\sqrt{2})(-5) = -5\sqrt{2}$
3. $(3)(\sqrt{2}) = 3\sqrt{2}$
4. $(3)(-5) = -15$

Step 3: Combine like terms

$2 - 5\sqrt{2} + 3\sqrt{2} - 15$

Combine the constants and the like terms involving $\sqrt{2}$:

1. Constants: $2 - 15 = -13$
2. Radicals: $-5\sqrt{2} + 3\sqrt{2} = -2\sqrt{2}$

Final Answer:

$-13 - 2\sqrt{2}$

Let me know if you want more details or have any questions!

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Here are five related questions to expand on this topic:

1. How do you simplify expressions with square roots?
2. What is the FOIL method, and when is it used?
3. Can you explain how to multiply two binomials with radicals in general terms?
4. How would the expression change if the second binomial was $(\sqrt{2} + 5)$ instead of $(\sqrt{2} - 5)$?
5. What happens if we square the binomial $(\sqrt{2} + 3)$ instead of multiplying it by $(\sqrt{2} - 5)$?

Tip:

Always group and simplify like terms (constants and radicals) to finalize your expression.