Let's expand the expression $(2 + 3\sqrt{3})^2$ using the formula for squaring a binomial:

$(a + b)^2 = a^2 + 2ab + b^2$

Here, $a = 2$ and $b = 3\sqrt{3}$. Applying the formula:

$(2 + 3\sqrt{3})^2 = 2^2 + 2(2)(3\sqrt{3}) + (3\sqrt{3})^2$

Now, calculate each term:

• $2^2 = 4$
• $2(2)(3\sqrt{3}) = 12\sqrt{3}$
• $(3\sqrt{3})^2 = 9 \times 3 = 27$

So, the expanded expression is:

$(2 + 3\sqrt{3})^2 = 4 + 12\sqrt{3} + 27$

Combine the constants:

$4 + 27 = 31$

Thus, the final answer is:

$(2 + 3\sqrt{3})^2 = 31 + 12\sqrt{3}$

Would you like to go over any step in more detail?

Here are some related questions:

1. How do you square a binomial with irrational numbers inside it?
2. What is the expansion formula for $(a + b)^2$?
3. How can you simplify expressions involving square roots?
4. What is the result when squaring a binomial with a negative term?
5. How do you handle multiplying square roots in algebra?

Tip: Always remember to expand binomials step by step to avoid mistakes, especially when working with square roots.