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To factor the expression \( 10x^2 + 6xy - 25xy - 15y^2 \) by grouping, let’s go through the steps:

1. **Group Terms**: Separate the terms into two groups:
   \[
   (10x^2 + 6xy) - (25xy + 15y^2)
   \]

2. **Factor Out the Common Factors**:
   - From the first group, \(10x^2 + 6xy\), we can factor out \(2x\):
     \[
     2x(5x + 3y)
     \]
   - From the second group, \( -25xy - 15y^2 \), we can factor out \(-5y\):
     \[
     -5y(5x + 3y)
     \]

   So, we now have:
   \[
   2x(5x + 3y) - 5y(5x + 3y)
   \]

3. **Factor Out the Common Binomial**:
   Now, we can factor out the common factor \(5x + 3y\):
   \[
   (5x + 3y)(2x - 5y)
   \]

Therefore, the fully factored form of the expression is:
\[
(5x + 3y)(2x - 5y)
\]

Would you like more details on any step, or have further questions?

Here are related questions to deepen your understanding:

1. How do you recognize when to use grouping in factoring?
2. What is the significance of factoring out the greatest common factor in each grouping?
3. Can you factor other expressions with four terms using similar techniques?
4. What would happen if the terms couldn’t be grouped with a common binomial factor?
5. Why is it helpful to rewrite expressions to simplify factoring?

**Tip:** When using grouping, always double-check that each grouped term shares a factor and that the remaining factors are identical. This ensures successful factoring.

The middle term of the trinomial has been rewritten. Now factor by grouping: 10x^2 + 6xy - 25xy - 15y^2

Explore how to factor the polynomial 10x^2 + 6xy - 25xy - 15y^2 by grouping. This solution covers essential algebraic techniques such as factoring by grouping and applying the distributive property. Ideal for high school students in Grades 9-12.

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