Factor by grouping.
ab plus 3 a minus 2 b minus 6

To factor the expression \(ab + 3a - 2b - 6\) by grouping, follow these steps:

1. **Group the terms**: Split the expression into two pairs of terms:
   \[
   (ab + 3a) - (2b + 6)
   \]

2. **Factor out the greatest common factor (GCF)** from each group:
   - In the first group \((ab + 3a)\), the GCF is \(a\):
     \[
     a(b + 3)
     \]
   - In the second group \((-2b - 6)\), the GCF is \(-2\) (be careful to factor out the negative sign):
     \[
     -2(b + 3)
     \]

3. **Factor out the common binomial factor**:
   - Both groups now have the common binomial factor \((b + 3)\):
     \[
     a(b + 3) - 2(b + 3)
     \]

4. **Combine the factors**:
   - Factor out the common binomial \((b + 3)\):
     \[
     (a - 2)(b + 3)
     \]

So, the factored form of the expression \(ab + 3a - 2b - 6\) is:
\[
(a - 2)(b + 3)
\]

Would you like any further explanation, or do you have any questions?

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Here are five related questions you might find interesting:
1. How would you factor the expression \(xy + 2x - 3y - 6\) by grouping?
2. What is the factored form of \(ac + 4a - 5c - 20\)?
3. How can you check that the factorization \((a - 2)(b + 3)\) is correct?
4. What strategies can be used if the terms don't initially seem to group nicely?
5. How do you factor expressions where the middle terms involve both variables, like \(ax + ay + bx + by\)?

**Tip:** When grouping terms, ensure the signs are handled correctly when factoring out the greatest common factor, especially with negative signs.

Learn how to factor expressions by grouping using a step-by-step method with an example. This problem covers key algebraic concepts such as factoring by grouping and is suitable for students in Grades 9-10.

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