To factor the expression:

$3ax - 12bc - 2a + 8b$

Step 1: Group the terms

Group the terms in pairs to facilitate factoring: $(3ax - 2a) - (12bc - 8b)$

Step 2: Factor out the greatest common factor (GCF) from each group

For the first group $(3ax - 2a)$, the GCF is $a$: $a(3x - 2)$

For the second group $-(12bc - 8b)$, the GCF is $-4b$: $-4b(3c - 2)$

The expression becomes: $a(3x - 2) - 4b(3c - 2)$

Step 3: Check for a common binomial factor

There is no common binomial factor ($3x - 2 \neq 3c - 2$). Therefore, the expression cannot be factored further into simpler terms.

Final Answer:

$a(3x - 2) - 4b(3c - 2)$

Let me know if you need further clarification or assistance!

Relative Questions:

1. How do you determine the greatest common factor (GCF) for terms in an expression?
2. Why do we group terms when factoring?
3. Can this expression be factored further in a specific case (e.g., if $x = c$)?
4. What strategies can be used if the expression involves more terms?
5. How does factoring help in solving equations?

Tip: Always double-check for common factors or patterns that might simplify your expression further.