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.

This step-by-step guide explains how to simplify the expression (√3 + √7)(√3 - √5) using the distributive property (FOIL method). Learn about square roots, binomial multiplication, and how to combine terms. Suitable for students in Grades 9-10.

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