The given expression to simplify is:

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

We can simplify this using the distributive property (FOIL method for binomials):

1. Multiply the first terms: $3 \times 5 = 15$
2. Multiply the outer terms: $3 \times 5\sqrt{2} = 15\sqrt{2}$
3. Multiply the inner terms: $\sqrt{2} \times 5 = 5\sqrt{2}$
4. Multiply the last terms: $\sqrt{2} \times 5\sqrt{2} = 5 \times (\sqrt{2})^2 = 5 \times 2 = 10$

Now combine the results:

$15 + 15\sqrt{2} + 5\sqrt{2} + 10$

Simplify the like terms:

$15 + 10 + (15\sqrt{2} + 5\sqrt{2}) = 25 + 20\sqrt{2}$

Thus, the simplified expression is:

$25 + 20\sqrt{2}$

Would you like more details on this, or have any questions?

Here are 5 related questions:

1. How do you apply the distributive property to binomials?
2. What are the rules for simplifying radical expressions?
3. How can you handle more complex expressions with multiple radicals?
4. What happens if negative quantities are involved in similar problems?
5. Can radicals always be simplified in this way?

Tip: When multiplying radicals, remember that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$, which helps simplify expressions like $\sqrt{2} \times \sqrt{2}$.