Let's go through each of these expressions one by one and factorize them.

Problem a

Expression: $xy + xz - wy - wz$

Factor by grouping: $= x(y + z) - w(y + z)$ Now factor out $y + z$: $= (y + z)(x - w)$

Solution: $(y + z)(x - w)$

Problem b

Expression: $ab + bc - ad - cd$

Factor by grouping: $= b(a + c) - d(a + c)$ Now factor out $a + c$: $= (a + c)(b - d)$

Solution: $(a + c)(b - d)$

Problem c

Expression: $5a + 15 - ab - 3b$

Factor by grouping: $= 5(a + 3) - b(a + 3)$ Now factor out $a + 3$: $= (a + 3)(5 - b)$

Solution: $(a + 3)(5 - b)$

Problem d

Expression: $6x - 24 - xy + 4y$

Factor by grouping: $= 6(x - 4) - y(x - 4)$ Now factor out $x - 4$: $= (x - 4)(6 - y)$

Solution: $(x - 4)(6 - y)$

Problem e

Expression: $11y + 22 - xy - 2x$

Factor by grouping: $= 11(y + 2) - x(y + 2)$ Now factor out $y + 2$: $= (y + 2)(11 - x)$

Solution: $(y + 2)(11 - x)$

Problem f

Expression: $ax^2 - ax - x + 1$

Factor by grouping: $= ax(x - 1) - 1(x - 1)$ Now factor out $x - 1$: $= (x - 1)(ax - 1)$

Solution: $(x - 1)(ax - 1)$

If you have any questions or would like further details on any specific factorization, feel free to ask!

Here are some related questions for further practice:

1. How would you factorize an expression with three terms (a trinomial) like $x^2 + 5x + 6$?
2. What technique would you use to factorize $x^2 - 4y^2$, which is a difference of squares?
3. How would you approach factorizing a cubic polynomial, such as $x^3 - 3x^2 + 3x - 1$?
4. What steps can help in identifying common factors in an expression?
5. How do you factor expressions where grouping does not immediately yield a common factor?

Tip: Always look for common factors first in each pair of terms before moving on to other techniques like grouping.