To factorize the expression \(12 - 2c - 6d + cd\), let's follow these steps:

1. **Group the terms**: Group the terms that share common factors.
   \[
   (12 - 6d) + (-2c + cd)
   \]

2. **Factor out the common factors**:
   - From \(12 - 6d\), you can factor out 6:
     \[
     6(2 - d)
     \]
   - From \(-2c + cd\), you can factor out \(c\):
     \[
     c(-2 + d)
     \]

3. **Rewrite the expression**:
   \[
   6(2 - d) + c(-2 + d)
   \]

4. **Observe a common factor**: Notice that \(2 - d\) and \(-2 + d\) are negatives of each other. We can rewrite \(c(-2 + d)\) as \(-c(2 - d)\), so we have:
   \[
   6(2 - d) - c(2 - d)
   \]

5. **Factor out the common term \((2 - d)\)**:
   \[
   (2 - d)(6 - c)
   \]

Thus, the factorized form of the expression \(12 - 2c - 6d + cd\) is:
\[
(2 - d)(6 - c)
\]

Would you like a detailed breakdown of any specific part of this process?

Here are 5 follow-up questions:
1. What is the significance of grouping terms when factorizing an expression?
2. How can you recognize common factors in more complex expressions?
3. Why do we rewrite terms like \(-2 + d\) as \(-c(2 - d)\)?
4. How can factoring help in solving algebraic equations?
5. What are some common mistakes to avoid when factoring expressions?

**Tip**: When factorizing, always check for negative signs and common factors before proceeding with more complex methods.

Learn how to factor the algebraic expression 12 - 2c - 6d + cd by grouping terms and using the distributive property. This step-by-step solution is suitable for students in Grades 9-11 and includes key algebraic concepts such as factoring common terms.

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