Let's simplify the expression $(12 - 5\sqrt{3})^2$.

To do this, we apply the formula for squaring a binomial: $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 12$ and $b = 5\sqrt{3}$.

Step 1: Square the first term

$a^2 = (12)^2 = 144$

Step 2: Multiply the two terms and double the result

$2ab = 2 \times 12 \times 5\sqrt{3} = 120\sqrt{3}$

Step 3: Square the second term

$b^2 = (5\sqrt{3})^2 = 25 \times 3 = 75$

Step 4: Combine all the terms

Now, we put everything together: $(12 - 5\sqrt{3})^2 = 144 - 120\sqrt{3} + 75$

Step 5: Simplify the result

$144 + 75 = 219$

So, the final simplified expression is: $(12 - 5\sqrt{3})^2 = 219 - 120\sqrt{3}$

Would you like any further explanation or have more questions about surds?

Related questions:

1. What is the conjugate of $12 - 5\sqrt{3}$?
2. How do you simplify the expression $(a + b)^2$ for surds?
3. What is the value of $(\sqrt{3} - 2)^2$?
4. How would you rationalize the denominator in an expression like $\frac{1}{\sqrt{2}}$?
5. Can you expand and simplify $(3 + \sqrt{5})^2$?

Tip: When dealing with binomials involving square roots, always apply the binomial expansion formula carefully and simplify each term step by step.