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To factor the quadratic expression \( 5x^2 + 17xy + 14y^2 \), we can follow these steps:

### Step-by-Step Factorization:

1. **Recognize the quadratic form:**
   The expression is of the form \( ax^2 + bxy + cy^2 \), where:
   - \( a = 5 \)
   - \( b = 17 \)
   - \( c = 14 \)

2. **Find two numbers that multiply to \( ac \) (the product of 5 and 14, which equals 70) and add to \( b = 17 \):**
   We need two numbers whose product is 70 and whose sum is 17. These numbers are 7 and 10, because:
   \[
   7 \times 10 = 70 \quad \text{and} \quad 7 + 10 = 17
   \]

3. **Rewrite the middle term using 7 and 10:**
   Break down \( 17xy \) into two terms:
   \[
   5x^2 + 7xy + 10xy + 14y^2
   \]

4. **Factor by grouping:**
   Group the terms in pairs:
   \[
   (5x^2 + 7xy) + (10xy + 14y^2)
   \]
   Factor out the greatest common factor (GCF) from each group:
   \[
   x(5x + 7y) + 2y(5x + 7y)
   \]

5. **Factor out the common binomial factor:**
   Both groups contain the factor \( (5x + 7y) \), so factor it out:
   \[
   (5x + 7y)(x + 2y)
   \]

### Final Factorized Form:
The factorization of \( 5x^2 + 17xy + 14y^2 \) is:
\[
(5x + 7y)(x + 2y)
\]

Let me know if you'd like further details or have any questions!

Here are 5 questions to expand this concept:
1. How do we identify when a quadratic expression can be factored?
2. What other methods can be used to factor quadratic expressions besides grouping?
3. Can all quadratic expressions be factored? If not, what alternatives exist?
4. How can the discriminant be useful when factoring quadratic equations?
5. What is the geometric interpretation of factoring quadratic expressions?

**Tip:** When factoring quadratics, always look for patterns, such as perfect square trinomials or differences of squares!

Factor the quadratic expression 5x^2 + 17xy + 14y^2.

Learn how to factor the quadratic expression 5x^2 + 17xy + 14y^2 step by step. This problem involves recognizing the quadratic form and using the grouping method for factorization. Suitable for students in Grades 9-10.

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