Question: Select the correct answer.
Which expression shows the factors of $9x^2 + 3x - 2$?
$(3x - 1)(3x - 2)$
$(3x + 1)(3x - 2)$
$(3x + 2)(3x - 1)$
$(9x - 2)(x + 1)$
1. **State the problem:** Factor the quadratic expression $$9x^2 + 3x - 2$$.
2. **Recall the factoring method:** For a quadratic $$ax^2 + bx + c$$, we look for two binomials $$(mx + n)(px + q)$$ such that:
- $$m imes p = a$$
- $$n imes q = c$$
- $$m imes q + n imes p = b$$
3. **Apply to our problem:** Here, $$a=9$$, $$b=3$$, and $$c=-2$$.
4. **Find factor pairs:**
- Factors of $$a=9$$ are $$(3,3)$$ or $$(9,1)$$.
- Factors of $$c=-2$$ are $$(1,-2), (-1,2), (2,-1), (-2,1)$$.
5. **Test combinations:**
- Try $$(3x + 2)(3x - 1)$$:
$$3x \times 3x = 9x^2$$
$$3x \times (-1) = -3x$$
$$2 \times 3x = 6x$$
$$2 \times (-1) = -2$$
Sum of middle terms: $$-3x + 6x = 3x$$, which matches the middle term.
6. **Verify the product:**
$$ (3x + 2)(3x - 1) = 9x^2 - 3x + 6x - 2 = 9x^2 + 3x - 2 $$
7. **Conclusion:** The correct factorization is $$(3x + 2)(3x - 1)$$.
**Final answer:** $(3x + 2)(3x - 1)$