Strategy: When I examine the given expression, 2x² - 2xy + y² - 2x + 3, I see that the expression "2x² - 2xy + y²" closely resembles the special product x² - 2xy + y², which conveniently factors to become (x - y)².
To see whether this special product is part of our solution, I'm going to explore whether there is any benefit to rewriting 2x² as x² + x².

Given: 2x² - 2xy + y² - 2x + 3
Rewrite as follows: x² + x² - 2xy + y² - 2x + 3
Rearrange to get: (x² - 2xy + y²) + x² - 2x + 3
Factor the bracketed expression to get: (x - y)² + x² - 2x + 3

Strategy: Now that I've expressed part of the expression as (x - y)², I recognize that x² - 2x + 3 looks a lot like x² - 2x + 1, which can be factored to get (x - 1)²

So, let's rewrite the expression as follows: (x - y)² + (x² - 2x + 1) + 2
Factor the bracketed expression to get: (x - y)² + (x - 1)² + 2

Important property: something² is always greater than or equal to 0.
So, the smallest possible value of (x - y)² is zero, and the smallest possible value of (x - 1)² is also 0.
In fact, when x = 1 and y = 1, (x - y)² = 0, and (x - 1)² = 0

Of course, the expression we want to minimize is (x - y)² + (x - 1)² + 2
So, when x = 1 and y = 1, we get: (x - y)² + (x - 1)² + 2 = (1 - 1)² + (1 - 1)² + 2 = 0² + 0² + 2 = 2

This means 2 is the smallest possible value of the given expression (and this minimum value occurs when x = y = 1)

Answer: C