To simplify the expression \((3\sqrt{5} + 5)(5\sqrt{3} + 3)\), we'll use the distributive property (also known as the FOIL method for binomials). This property states that:

\[
(a + b)(c + d) = ac + ad + bc + bd
\]

Here, let:
\( a = 3\sqrt{5} \)
\( b = 5 \)
\( c = 5\sqrt{3} \)
\( d = 3 \)

Now, applying the distributive property:

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

Calculate each term:

1. \( (3\sqrt{5})(5\sqrt{3}) = 3 \cdot 5 \cdot \sqrt{5 \cdot 3} = 15\sqrt{15} \)
2. \( (3\sqrt{5})(3) = 9\sqrt{5} \)
3. \( (5)(5\sqrt{3}) = 25\sqrt{3} \)
4. \( (5)(3) = 15 \)

Combine all the terms:

\[
15\sqrt{15} + 9\sqrt{5} + 25\sqrt{3} + 15
\]

So, the simplified form of \((3\sqrt{5} + 5)(5\sqrt{3} + 3)\) is:

\[
15\sqrt{15} + 9\sqrt{5} + 25\sqrt{3} + 15
\]

Would you like a detailed explanation or have any questions?

Here are some related questions:

1. How would you simplify \((2\sqrt{7} + 4)(3\sqrt{2} + 1)\)?
2. What is the distributive property and how is it used in algebra?
3. Can you simplify \((4\sqrt{6} + 2)(2\sqrt{3} + 5)\)?
4. How do you handle binomials with radicals in general?
5. How do you simplify \((\sqrt{2} + \sqrt{3})(\sqrt{5} + \sqrt{7})\)?
6. What are some common mistakes to avoid when simplifying expressions with radicals?
7. How would you simplify \((\sqrt{a} + b)(\sqrt{c} + d)\) for general \(a, b, c,\) and \(d\)?
8. What is the result of \((\sqrt{3} + 2\sqrt{2})(\sqrt{3} - 2\sqrt{2})\)?

**Tip:** Always verify your final result by checking each step of the calculation, especially when dealing with radicals and binomials.

Learn how to simplify the expression (3√5 + 5)(5√3 + 3) using the distributive property (FOIL method). This problem involves radicals and demonstrates the application of algebraic techniques to simplify expressions. Suitable for high school students.

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