The given equation is:

$-1 + 2a + 2b - a^2 - b^2 + 2ab = 0$

Step 1: Rearrange the terms

We can rearrange the equation to group similar terms together:

$a^2 + b^2 - 2ab - 2a - 2b + 1 = 0$

This form is now easier to analyze.

Step 2: Factor the quadratic terms

Next, we aim to simplify the equation. Recognize that the terms involving $a^2$, $b^2$, and $ab$ resemble a perfect square expression. In fact, the first three terms can be factored as:

$(a - b)^2$

So the equation becomes:

$(a - b)^2 - 2a - 2b + 1 = 0$

Step 3: Simplify the equation

Now we solve the equation for $a$ and $b$. We rewrite it as:

$(a - b)^2 = 2a + 2b - 1$

At this point, solving for specific values of $a$ and $b$ might require trial and error or further manipulation, depending on the conditions given. If you have any constraints on $a$ or $b$, I can proceed with further steps.

Would you like to continue with this process or clarify any conditions on $a$ and $b$? Here are some related questions:

1. Do you have any specific values for either $a$ or $b$?
2. Would you like to explore numerical methods for solving this?
3. Should we try a different approach, like substituting specific values?
4. Would you like a graphical representation of the solution?
5. Do you want to analyze particular cases, such as when $a = b$?

Tip: When solving non-linear equations, sometimes factoring or completing the square reveals useful patterns.