Let's go through the expression step by step to ensure everything is correct:

Given expression: $(\sqrt{2} + \sqrt{12})(\sqrt{18} + 3\sqrt{3})$

Step 1: Simplify each square root

$\sqrt{12} = 2\sqrt{3}, \quad \sqrt{18} = 3\sqrt{2}$ So the expression becomes: $(\sqrt{2} + 2\sqrt{3})(3\sqrt{2} + 3\sqrt{3})$

Step 2: Apply the distributive property (FOIL method)

Multiply each term in the first binomial by each term in the second binomial: $\sqrt{2} \times 3\sqrt{2} + \sqrt{2} \times 3\sqrt{3} + 2\sqrt{3} \times 3\sqrt{2} + 2\sqrt{3} \times 3\sqrt{3}$

Step 3: Simplify each multiplication

1. $\sqrt{2} \times 3\sqrt{2} = 3 \times \sqrt{2 \times 2} = 3 \times \sqrt{4} = 3 \times 2 = 6$
2. $\sqrt{2} \times 3\sqrt{3} = 3\sqrt{6}$
3. $2\sqrt{3} \times 3\sqrt{2} = 6\sqrt{6}$
4. $2\sqrt{3} \times 3\sqrt{3} = 6 \times \sqrt{9} = 6 \times 3 = 18$

Step 4: Combine like terms

$6 + 3\sqrt{6} + 6\sqrt{6} + 18$

Combine the terms involving $\sqrt{6}$: $6 + 18 + 3\sqrt{6} + 6\sqrt{6} = 24 + 9\sqrt{6}$

So, the simplified expression is: $(\sqrt{2} + \sqrt{12})(\sqrt{18} + 3\sqrt{3}) = 24 + 9\sqrt{6}$

This confirms that your solution is correct!

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Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How does the FOIL method change when dealing with complex numbers instead of real numbers?
2. What happens if you multiply two conjugate binomials containing square roots?
3. How can you simplify the expression if one of the terms inside the square root is negative?
4. What is the geometric interpretation of multiplying binomials with radicals?
5. How would you approach simplifying expressions with cube roots in a similar way?

Tip: Always double-check each step when simplifying expressions, especially when dealing with radicals, to avoid common errors.