To factor the quadratic function \( f(x) = 5x^2 + 9x - 2 \), we'll use the method of splitting the middle term or applying the quadratic formula to find the roots and express the quadratic in its factored form.

The equation is of the form \( ax^2 + bx + c \), where:
- \( a = 5 \)
- \( b = 9 \)
- \( c = -2 \)

We need to find two numbers that multiply to \( ac = 5 \times -2 = -10 \) and add up to \( b = 9 \).

These numbers are 10 and -1, because:
- \( 10 \times (-1) = -10 \)
- \( 10 + (-1) = 9 \)

Now, rewrite the middle term \( 9x \) as \( 10x - x \):
\[
5x^2 + 10x - x - 2
\]

Next, group the terms:
\[
(5x^2 + 10x) - (x + 2)
\]

Factor out the greatest common factor from each group:
\[
5x(x + 2) - 1(x + 2)
\]

Now, factor out the common binomial factor \( (x + 2) \):
\[
(5x - 1)(x + 2)
\]

Thus, the factored form of \( f(x) = 5x^2 + 9x - 2 \) is:
\[
f(x) = (5x - 1)(x + 2)
\]

Let me know if you need further details or explanations! Here are some related questions:

1. How do you factor a quadratic when it doesn't have real roots?
2. Can you factor using the quadratic formula approach?
3. How do you handle quadratics with negative coefficients?
4. What are some other methods to factor polynomials?
5. How do you graph a quadratic function once it's factored?

**Tip**: Always check your factorization by expanding back to ensure it matches the original equation!

Learn how to factor the quadratic function f(x) = 5x^2 + 9x - 2 using the method of splitting the middle term. This detailed step-by-step solution covers key algebraic concepts such as factoring by grouping and the Zero Product Property. Suitable for students in Grades 9-11.

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