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We need to expand and fully simplify \((5 + \sqrt{3})(1 + \sqrt{3})\).  

### Step 1: Expand using the distributive property (FOIL method)  
\[
(5 + \sqrt{3})(1 + \sqrt{3}) = 5(1) + 5(\sqrt{3}) + \sqrt{3}(1) + \sqrt{3}(\sqrt{3})
\]

### Step 2: Multiply each term  
\[
= 5 + 5\sqrt{3} + \sqrt{3} + (\sqrt{3})^2
\]

Since \((\sqrt{3})^2 = 3\), we replace it in the equation:

\[
= 5 + 5\sqrt{3} + \sqrt{3} + 3
\]

### Step 3: Simplify the expression  
Combine like terms:

\[
(5 + 3) + (5\sqrt{3} + \sqrt{3})
\]

\[
= 8 + 6\sqrt{3}
\]

### Final Answer:
\[
\mathbf{8 + 6\sqrt{3}}
\]

Would you like me to explain any step further?  

#### Related Questions:
1. How do you apply the FOIL method to binomial multiplication?
2. Why does \((\sqrt{3})^2 = 3\)?
3. How would the expansion change if the terms were different, e.g., \((a + b)(c + d)\)?
4. What are some common mistakes students make when simplifying expressions with square roots?
5. How do you rationalize the denominator in expressions like \(\frac{1}{1+\sqrt{3}}\)?

#### Math Tip:
Always combine like terms carefully when simplifying expressions, especially when working with radicals!

Expand and fully simplify (5 + √3)(1 + √3)

Learn how to expand and simplify the expression (5 + √3)(1 + √3) using the distributive property and radical arithmetic. Perfect for students in Grades 8-10, this example explains each step of the binomial expansion and radical simplification.

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