To factor the quadratic function \( f(x) = 5x^2 + 9x - 2 \), we need to find two numbers that multiply to \( 5 \times (-2) = -10 \) and add up to \( 9 \).

### Step-by-step Solution:

1. **Identify the Coefficients:**
   The quadratic is in the form \( ax^2 + bx + c \), where:
   \[
   a = 5, \quad b = 9, \quad c = -2
   \]

2. **Find Two Numbers that Multiply to \( ac = -10 \) and Add to \( b = 9 \):**
   We are looking for two numbers that multiply to \(-10\) and add to \(9\). The numbers are \(10\) and \(-1\).

3. **Rewrite the Middle Term Using These Two Numbers:**
   Rewrite \(9x\) as \(10x - x\):
   \[
   5x^2 + 10x - x - 2
   \]

4. **Factor by Grouping:**
   Group the terms in pairs:
   \[
   (5x^2 + 10x) - (x + 2)
   \]
   Factor out the common factors in each group:
   \[
   5x(x + 2) - 1(x + 2)
   \]

5. **Factor Out the Common Binomial:**
   Notice that \((x + 2)\) is common in both terms:
   \[
   (5x - 1)(x + 2)
   \]

### Conclusion:

The factored form of \( f(x) = 5x^2 + 9x - 2 \) is:

\[
\boxed{(5x - 1)(x + 2)}
\]

Would you like more details on the factoring process or have any specific questions?  

Here are some related questions you might consider:  

1. How do you factor quadratic expressions when the leading coefficient is greater than 1?  
2. What is the significance of factoring in solving quadratic equations?  
3. How can you verify the correctness of a factored quadratic?  
4. What methods can be used if factoring is challenging or not obvious?  
5. How does the quadratic formula relate to factoring?  

**Tip:** When factoring quadratics, always check your solution by expanding the factors to see if you get the original expression back!  

Learn how to factor the quadratic equation 5x^2 + 9x - 2 using factoring by grouping. This solution includes key algebraic concepts, such as identifying coefficients and applying the Zero Product Property. Suitable for students in Grades 9-10.

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