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.