https://file.aiquickdraw.com/mathbot/file/173062994168465nigvw7.jpg

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.

Factorise the following expressions: a) xy + xz - wy - wz, b) ab + bc - ad - cd, c) 5a + 15 - ab - 3b, d) 6x - 24 - xy + 4y, e) 11y + 22 - xy - 2x, f) ax^2 - ax - x + 1

Learn how to factorise various algebraic expressions using the grouping method. This problem covers algebraic concepts such as factorisation and the distributive property, suitable for students in Grades 8-10.

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