The given expression can be written as:

x^2 + y^2 - 2xy + 2x - 2y

We can rearrange the terms to group the x and y terms together:

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

Now, let's complete the square for the x terms:

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

The first three terms in the parentheses can be factored as a perfect square:

(x - y)^2 + (2x - y^2)

Similarly, let's complete the square for the y terms:

(y^2 - 2y) = (y^2 - 2y + 1) - 1

The first three terms in the parentheses can be factored as a perfect square:

(y - 1)^2 - 1

Now, we can substitute these expressions back into the original equation:

(x - y)^2 + (2x - y^2) + (y - 1)^2 - 1

Simplifying further, we get:

(x - y)^2 + (y - 1)^2 + (2x - y^2) - 1

Therefore, the expression x^2 + y^2 - 2xy + 2x - 2y can be written as (x - y)^2 + (y - 1)^2 + (2x - y^2) - 1.