The expression shown in the image is:

$(a - a\sqrt{2})^2$

We can simplify this by expanding the square using the binomial expansion formula:

$(x - y)^2 = x^2 - 2xy + y^2$

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

$(a - a\sqrt{2})^2 = a^2 - 2(a)(a\sqrt{2}) + (a\sqrt{2})^2$

Now simplify each term:

1. $a^2$ stays as is.
2. $-2(a)(a\sqrt{2}) = -2a^2\sqrt{2}$.
3. $(a\sqrt{2})^2 = a^2 \cdot 2 = 2a^2$.

Putting it all together:

$a^2 - 2a^2\sqrt{2} + 2a^2$

This is the fully simplified form:

$a^2 - 2a^2\sqrt{2} + 2a^2$

Let me know if you need any further details or clarifications!

Here are 5 related questions to explore:

1. How would the expansion change if the square root was $\sqrt{3}$ instead of $\sqrt{2}$?
2. Can you apply this binomial expansion to a cubic expression like $(a - a\sqrt{2})^3$?
3. What happens if you replace $a$ with a specific value, like 1 or 2, and simplify further?
4. How would the result change if the sign in the original expression was positive, i.e., $(a + a\sqrt{2})^2$?
5. How is the binomial expansion used in real-world applications, such as physics or engineering?

Tip: Binomial expansions become very helpful in algebra when dealing with powers of binomials. Mastering this technique will aid in solving more complex equations easily.