You're right; I made a mistake in my earlier response. Let's go through the correct steps to factor the trinomial \(9x^2 + 27x + 14\).

### Step-by-Step Solution:

We need to factor the trinomial:

\[
9x^2 + 27x + 14
\]

1. **Identify coefficients:**
   - \(a = 9\) (coefficient of \(x^2\))
   - \(b = 27\) (coefficient of \(x\))
   - \(c = 14\) (constant term)

2. **Find two numbers that multiply to \(ac = 9 \times 14 = 126\) and add up to \(b = 27\):**
   - The factors of 126 are \(1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126\).

   We need two numbers that multiply to 126 and add up to 27. The correct numbers are **9 and 14** since:
   \[
   9 \times 14 = 126 \quad \text{and} \quad 9 + 14 = 27.
   \]

3. **Rewrite the middle term using these two numbers:**

   Split the middle term, \(27x\), into \(9x + 18x\):

   \[
   9x^2 + 9x + 18x + 14
   \]

4. **Group the terms and factor by grouping:**

   Group the terms:

   \[
   (9x^2 + 9x) + (18x + 14)
   \]

   Factor out the greatest common factor from each group:

   \[
   9x(x + 1) + 14(x + 1)
   \]

   Notice that we have a common factor of \((x + 1)\):

   \[
   (9x + 14)(x + 1)
   \]

### Conclusion:

The correct factored form of the trinomial \(9x^2 + 27x + 14\) is:

\[
\boxed{(9x + 14)(x + 1)}
\]

Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. How can you verify if a factored form is correct?
2. What are the differences between factoring by grouping and using the quadratic formula?
3. How do you handle trinomials with negative coefficients?
4. What are the applications of factoring in solving quadratic equations?
5. Can all trinomials be factored over the set of real numbers?

**Tip:** When factoring, always double-check the sum and product of the numbers chosen to ensure they match the coefficients correctly.

Learn how to factor the trinomial 9x^2 + 27x + 14 step-by-step with a detailed solution. This problem covers key algebraic concepts, including factoring trinomials and identifying correct factors. Suitable for high school students.

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