To solve \((5 + \sqrt{5})(2 + \sqrt{5})\), we can use the distributive property (also known as the FOIL method for binomials):

\[
(5 + \sqrt{5})(2 + \sqrt{5}) = 5 \cdot 2 + 5 \cdot \sqrt{5} + \sqrt{5} \cdot 2 + \sqrt{5} \cdot \sqrt{5}
\]

Now, calculate each term:

1. \(5 \cdot 2 = 10\)
2. \(5 \cdot \sqrt{5} = 5\sqrt{5}\)
3. \(\sqrt{5} \cdot 2 = 2\sqrt{5}\)
4. \(\sqrt{5} \cdot \sqrt{5} = 5\)

Now, combine all the terms:

\[
10 + 5\sqrt{5} + 2\sqrt{5} + 5
\]

Simplify:

\[
(10 + 5) + (5\sqrt{5} + 2\sqrt{5}) = 15 + 7\sqrt{5}
\]

Thus, the result is:

\[
15 + 7\sqrt{5}
\]

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:
1. What is the general rule for multiplying binomials?
2. How would you multiply \( (a + \sqrt{b})(c + \sqrt{d}) \)?
3. What happens when you multiply two conjugates like \( (a + \sqrt{b})(a - \sqrt{b}) \)?
4. How do you simplify expressions involving square roots?
5. How would you apply the distributive property to three binomials?

**Tip:** When multiplying terms with square roots, remember that \( \sqrt{a} \times \sqrt{a} = a \).

Learn how to multiply the binomials (5+√5)(2+√5) using the distributive property and simplify the expression to obtain 15 + 7√5. This problem covers key algebraic concepts and properties of square roots, suitable for students in Grades 8-10.

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