To factor by grouping, we'll first organize the terms and then group them in pairs:

### Expression:  
\[ x^2y - 6y - 7x^2 + 42 \]

1. **Group the terms**:  
   Group terms that have common factors:  
   \[ (x^2y - 6y) + (-7x^2 + 42) \]

2. **Factor out the greatest common factor (GCF) from each group**:
   - From \( x^2y - 6y \), factor out \( y \):  
     \[ y(x^2 - 6) \]
   - From \( -7x^2 + 42 \), factor out \( -7 \):  
     \[ -7(x^2 - 6) \]

Now, the expression looks like this:  
\[ y(x^2 - 6) - 7(x^2 - 6) \]

3. **Factor out the common binomial \( (x^2 - 6) \)**:  
   \[ (x^2 - 6)(y - 7) \]

So, the factored form is:  
\[ (x^2 - 6)(y - 7) \]

Would you like further details or have any questions?

Here are 5 related questions you might explore:
1. How do you factor other quadratic expressions using grouping?
2. What is the process to factor a difference of squares, like \(x^2 - 6\)?
3. Can all quadratic expressions be factored by grouping?
4. What if the middle terms don't have a clear grouping pattern—what should you do?
5. How do you apply factorization techniques to solve equations?

**Tip:** Always look for the greatest common factor (GCF) first before attempting to factor further.

Factor by grouping: x^2y - 6y - 7x^2 + 42

Learn how to factor the expression x^2y - 6y - 7x^2 + 42 by grouping and finding the greatest common factor (GCF). This problem covers important algebraic techniques, such as factoring by grouping and recognizing common binomials. Suitable for students in Grades 9-11.

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