The expression provided is:

$\left( \sqrt{6 - 2\sqrt{5}} - \sqrt{6 + 2\sqrt{5}} \right)^2$

Let's simplify this step by step:

Step 1: Apply the formula for $(a - b)^2$:

$(a - b)^2 = a^2 - 2ab + b^2$ Here, $a = \sqrt{6 - 2\sqrt{5}}$ and $b = \sqrt{6 + 2\sqrt{5}}$.

Step 2: Calculate $a^2$ and $b^2$:

$a^2 = 6 - 2\sqrt{5}, \quad b^2 = 6 + 2\sqrt{5}$

Step 3: Add $a^2$ and $b^2$:

$a^2 + b^2 = (6 - 2\sqrt{5}) + (6 + 2\sqrt{5}) = 12$

Step 4: Calculate the product $2ab$:

$ab = \sqrt{(6 - 2\sqrt{5})(6 + 2\sqrt{5})}$ This is a difference of squares: $ab = \sqrt{(6)^2 - (2\sqrt{5})^2} = \sqrt{36 - 20} = \sqrt{16} = 4$ So, $2ab = 2 \times 4 = 8$.

Step 5: Putting it all together:

Now, substitute back into the expansion: $(a - b)^2 = a^2 + b^2 - 2ab = 12 - 8 = 4$

Thus, the simplified result is: $\boxed{4}$

Let me know if you'd like more details or have any questions!

Here are some related questions:

1. What are the properties of square roots that allow us to simplify radical expressions?
2. How does the difference of squares formula help in solving radical expressions like these?
3. Why is the value of $ab$ simplified as a perfect square in this problem?
4. How would this expression change if the signs inside the square roots were different?
5. What role does symmetry play in simplifying this kind of expression?

Tip: Always look for opportunities to apply the difference of squares formula when dealing with terms like $a - b$ where both involve square roots!