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To factor \(48xy + 72x^2 + 56y - 84x\) by grouping, let's break down the expression and group terms.

Given expression:
\[
48xy + 72x^2 + 56y - 84x
\]

1. **Step 1: Group terms.**
   \[
   (48xy + 72x^2) + (56y - 84x)
   \]

2. **Step 2: Factor out the common terms from each group.**
   - In the first group, \(48xy + 72x^2\), we can factor out \(24x\):
     \[
     24x(2y + 3x)
     \]
   - In the second group, \(56y - 84x\), we can factor out \(28\):
     \[
     28(2y - 3x)
     \]

   So, we rewrite the expression as:
   \[
   24x(2y + 3x) + 28(2y - 3x)
   \]

3. **Step 3: Recognize a common factor.**
   The expression can now be factored as:
   \[
   4(6x + 7)(2y - 3x)
   \]

The correct answer is:
\[
\boxed{4(6x + 7)(2y - 3x)}
\]

It looks like option 3 (which is marked) is the correct one.

Would you like further details on this process?

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Here are 5 related questions to expand your understanding:
1. How can you apply factoring by grouping to a four-term polynomial in general?
2. Why is it important to first group terms with common factors?
3. What are other methods to factor if grouping does not work?
4. How can you check your answer after factoring to ensure it's correct?
5. What are real-life applications of factoring polynomials?

**Tip:** Always check each term after factoring to confirm that the factors multiply back to the original expression.

Factor by grouping: 48xy + 72x^2 + 56y - 84x

Learn how to factor the polynomial expression 48xy + 72x^2 + 56y - 84x by grouping. This step-by-step guide demonstrates factoring by grouping and the application of the distributive property, suitable for Grades 9-10 students.

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