Let's solve the expression $(\sqrt{3} + \sqrt{7})(\sqrt{3} - \sqrt{5})$.

We can apply the distributive property (also known as FOIL when applied to binomials):

$(\sqrt{3} + \sqrt{7})(\sqrt{3} - \sqrt{5}) = (\sqrt{3})(\sqrt{3}) + (\sqrt{3})(-\sqrt{5}) + (\sqrt{7})(\sqrt{3}) + (\sqrt{7})(-\sqrt{5})$

Step-by-step:

1. Multiply $\sqrt{3} \times \sqrt{3} = 3$.
2. Multiply $\sqrt{3} \times -\sqrt{5} = -\sqrt{15}$.
3. Multiply $\sqrt{7} \times \sqrt{3} = \sqrt{21}$.
4. Multiply $\sqrt{7} \times -\sqrt{5} = -\sqrt{35}$.

Now, combine all the terms:

$3 - \sqrt{15} + \sqrt{21} - \sqrt{35}$

Thus, the simplified expression is:

$3 - \sqrt{15} + \sqrt{21} - \sqrt{35}$

This is the final result.

Would you like further details or clarification on any of the steps?

Here are some related questions:

1. What is the distributive property in algebra?
2. How do we simplify expressions involving square roots?
3. Can we further simplify square roots like $\sqrt{15}$?
4. What happens if the original expression contains complex numbers?
5. How does FOIL work with more complex binomials?

Tip: Always double-check each term in a product to ensure correct application of properties.