Let's factor the given expression by grouping. The expression is:

$xy - x + 2y - 2$

1. Group the terms in pairs: $(xy - x) + (2y - 2)$

2. Factor out the common factor from each group: $x(y - 1) + 2(y - 1)$

3. Notice that $(y - 1)$ is a common factor. So, factor $(y - 1)$ out: $(x + 2)(y - 1)$

Thus, the expression can be factored as:

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

So, the correct choice is:

A. $xy - x + 2y - 2 = (x + 2)(y - 1)$

Would you like more details on the steps? Here are 5 related questions:

1. What is the significance of the common factor when grouping terms?
2. How can you verify that your factorization is correct?
3. Are there situations where grouping might not work?
4. How does the sign of each term affect the grouping process?
5. How do you factor expressions with more than four terms?

Tip: Always recheck your factorization by expanding the factors to ensure you return to the original expression.