To solve the expression \((3\sqrt{5} + 5)(5\sqrt{3} + 3)\), we will use the distributive property (also known as the FOIL method in the case of binomials). This involves multiplying each term in the first binomial by each term in the second binomial:

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

1. **First terms:** \(3\sqrt{5} \times 5\sqrt{3}\)
   \[
   3\sqrt{5} \times 5\sqrt{3} = 15\sqrt{15}
   \]

2. **Outer terms:** \(3\sqrt{5} \times 3\)
   \[
   3\sqrt{5} \times 3 = 9\sqrt{5}
   \]

3. **Inner terms:** \(5 \times 5\sqrt{3}\)
   \[
   5 \times 5\sqrt{3} = 25\sqrt{3}
   \]

4. **Last terms:** \(5 \times 3\)
   \[
   5 \times 3 = 15
   \]

Now, combine all these results:

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

So, the expanded form of the expression is:

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

Would you like to explore any specific steps in more detail or have any further questions?

Here are some related questions you might find useful:

1. How do you simplify expressions with square roots?
2. What is the FOIL method, and when is it used?
3. How do you multiply binomials with square roots?
4. How do you factor expressions involving square roots?
5. What are some common mistakes when multiplying binomials?
6. How do you identify like terms involving square roots?
7. Can square roots be simplified further in some cases?
8. What are the properties of square roots in algebra?

**Tip:** Always double-check the signs and coefficients when using the distributive property, as it's easy to make mistakes with square roots and large numbers!

Learn how to expand and solve the expression (3√5 + 5)(5√3 + 3) step-by-step using the FOIL method. This problem involves algebraic manipulation and the multiplication of binomials with square roots. Suitable for high school students studying algebra and binomial expansion.

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