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The problem you're asking for involves simplifying the expression:

\[
(\sqrt{5} + 5\sqrt{2})(3\sqrt{7} + 2\sqrt{2})
\]

We will simplify this by using the distributive property (also known as FOIL in this case).

Let's proceed step by step:

1. **Distribute** \( \sqrt{5} \) to both terms in the second bracket:
   \[
   \sqrt{5} \cdot 3\sqrt{7} + \sqrt{5} \cdot 2\sqrt{2}
   = 3\sqrt{35} + 2\sqrt{10}
   \]

2. **Distribute** \( 5\sqrt{2} \) to both terms in the second bracket:
   \[
   5\sqrt{2} \cdot 3\sqrt{7} + 5\sqrt{2} \cdot 2\sqrt{2}
   = 15\sqrt{14} + 10\cdot2 = 15\sqrt{14} + 20
   \]

Now, combine all the terms:

\[
3\sqrt{35} + 2\sqrt{10} + 15\sqrt{14} + 20
\]

This is the simplified form of the expression.

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Do you need further details on the steps or have any questions? 

Here are 5 related questions you can explore:
1. How do you multiply square roots in algebra?
2. What are common techniques for simplifying radical expressions?
3. How do you recognize like terms when simplifying expressions involving radicals?
4. Can you apply the distributive property to binomials with square roots?
5. What are the general steps to simplify an expression like this?

**Tip:** Always try to simplify radicals as much as possible before multiplying or adding them to see if there are common terms to combine.

Learn how to simplify the expression (√5 + 5√2)(3√7 + 2√2) using the distributive property and properties of square roots. This problem is suitable for students in Grades 10-12, covering key algebraic concepts and simplification techniques.

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