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.