The task is to factorize the given expressions. I’ll go through each of these factorization problems one by one.

1. $3x\omega - 3\omega y - (y - x)^2$

   To factorize this, observe if common factors can be extracted or if it's a difference of squares.

2. $\alpha^2 - \beta + \alpha - \alpha \beta$

   Here, group terms to factor out common factors. Possible grouping could simplify this into a product of two binomials.

3. $2x^2 - 2xy - 6 \alpha x + 6 \alpha y$

   Notice the terms involve both $x$ and $y$, as well as constants. Group terms containing $x$ and those with $y$ and factor accordingly.

4. $\alpha x^2 - \beta x^2 + x^2 + \alpha - \beta + 1$

   This can be simplified by factoring out $x^2$ from the terms containing it, followed by combining constants.

5. $( \alpha x + \beta y )^2 + ( \alpha y - \beta x )^2$

   Recognize this expression as a sum of squares. This might simplify using known identities for the sum of squares.

6. $2x \sqrt{xy} + x \sqrt{y} + y \sqrt{xy} + 2y \sqrt{xy}$, with $x, y > 0$

   Factor out common square roots and terms to achieve a simplified form.

7. $\alpha x^2 + 2 \alpha xy + \beta x + 2 \beta y - \beta x^2 - 2 \beta xy - \alpha x - 2 \alpha y$

   Group terms with $x^2$, $xy$, $x$, and constants to find common factors or combine like terms for simplification.

Would you like me to go through the detailed solution for each part?

Here are 5 questions to deepen understanding:

1. How can we recognize when to use grouping in factorization?
2. What are the common methods for factoring polynomials with multiple variables?
3. How can factoring by grouping help in simplifying expressions?
4. How do we approach factorization when square roots are involved?
5. What techniques are useful for factorizing expressions with sums of squares?

Tip: Always start by looking for the greatest common factor (GCF) before attempting more complex factorization techniques.