To simplify the expression \((3\sqrt{5} + 5)(5\sqrt{3} + 3)\), we need to use the distributive property (also known as the FOIL method for binomials), which stands for First, Outer, Inner, Last:

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

1. **First**: Multiply the first terms in each binomial: \(3\sqrt{5} \cdot 5\sqrt{3}\)
\[
3\sqrt{5} \cdot 5\sqrt{3} = 15\sqrt{15}
\]

2. **Outer**: Multiply the outer terms in the binomials: \(3\sqrt{5} \cdot 3\)
\[
3\sqrt{5} \cdot 3 = 9\sqrt{5}
\]

3. **Inner**: Multiply the inner terms in the binomials: \(5 \cdot 5\sqrt{3}\)
\[
5 \cdot 5\sqrt{3} = 25\sqrt{3}
\]

4. **Last**: Multiply the last terms in each binomial: \(5 \cdot 3\)
\[
5 \cdot 3 = 15
\]

Now, we combine all the terms:
\[
15\sqrt{15} + 9\sqrt{5} + 25\sqrt{3} + 15
\]

The expression \((3\sqrt{5} + 5)(5\sqrt{3} + 3)\) simplifies to:
\[
15\sqrt{15} + 9\sqrt{5} + 25\sqrt{3} + 15
\]

Would you like further details on any part of this solution? Do you have any other questions?

Here are 8 related questions for further practice:
1. Simplify \((2\sqrt{2} + 4)(3\sqrt{2} + 5)\).
2. Expand and simplify \((\sqrt{6} + 2)(\sqrt{6} - 2)\).
3. Simplify \((\sqrt{3} + 1)^2\).
4. Multiply and simplify \((4 + \sqrt{7})(4 - \sqrt{7})\).
5. Expand \((5\sqrt{2} + 1)(\sqrt{3} + 2)\).
6. Simplify \((\sqrt{8} + \sqrt{2})(\sqrt{8} - \sqrt{2})\).
7. Expand \((a\sqrt{b} + c)(d\sqrt{e} + f)\).
8. Simplify \((\sqrt{x} + y)(\sqrt{y} + x)\).

**Tip:** When multiplying binomials involving radicals, treat the radicals as variables and use the distributive property carefully to ensure all terms are correctly simplified.

Learn how to simplify the expression (3√5 + 5)(5√3 + 3) using the distributive property. This problem involves algebraic manipulation with radicals. Suitable for high school students.

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