Let's simplify the expression $(\sqrt{3} - 1)(\sqrt{3} + 5)$.

This is a product of two binomials, so we can apply the distributive property (also known as the FOIL method):

$(\sqrt{3} - 1)(\sqrt{3} + 5) = \sqrt{3} \cdot \sqrt{3} + \sqrt{3} \cdot 5 - 1 \cdot \sqrt{3} - 1 \cdot 5$

Now, let's calculate each part:

1. $\sqrt{3} \cdot \sqrt{3} = 3$
2. $\sqrt{3} \cdot 5 = 5\sqrt{3}$
3. $-1 \cdot \sqrt{3} = -\sqrt{3}$
4. $-1 \cdot 5 = -5$

Now, combine all the terms:

$3 + 5\sqrt{3} - \sqrt{3} - 5$

Simplify by combining like terms:

$(3 - 5) + (5\sqrt{3} - \sqrt{3}) = -2 + 4\sqrt{3}$

So, the simplified expression is:

$\boxed{-2 + 4\sqrt{3}}$

Would you like more details on this? Here are five related questions to explore:

1. How does the distributive property apply to complex numbers?
2. Can you explain how to simplify other binomial expressions involving square roots?
3. What are the steps for rationalizing binomial expressions?
4. How do you handle higher-order roots in algebraic expressions?
5. What is the importance of the FOIL method in simplifying algebraic expressions?

Tip: Always combine like terms carefully when simplifying, especially when dealing with radicals or irrational numbers.