You have the equation:

$x^2 - 2xy + y^2 + 2\sqrt{2}x + \sqrt{2}y = 0$

This looks like a quadratic equation in terms of $x$ and $y$. To solve or simplify it, we can attempt to complete the square or apply substitutions to potentially make it easier to solve.

Step 1: Grouping terms by $x$ and $y$

We have: $(x^2 - 2xy + y^2) + 2\sqrt{2}x + \sqrt{2}y = 0$

Notice that the first part $x^2 - 2xy + y^2$ can be rewritten as $(x - y)^2$, so the equation becomes:

$(x - y)^2 + 2\sqrt{2}x + \sqrt{2}y = 0$

Step 2: Try to simplify further

Now the equation looks like: $(x - y)^2 + \sqrt{2}(2x + y) = 0$

We can explore solutions by substituting possible values for $x$ and $y$ that satisfy this equation. Alternatively, if you're looking to solve for one variable in terms of the other, we can proceed further with that approach.

Would you like to continue with this solution, or explore specific values for $x$ and $y$? Let me know how you'd like to proceed!