Factor the​ trinomial, or state that the trinomial is prime.

9 x squared plus 27 x plus 149x2+27x+14

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Part 1

Select the correct choice below​ and, if​ necessary, fill in any answer boxes within your choice.

To factor the trinomial \(9x^2 + 27x + 14\), we'll first check if it can be factored into the form:

\[
(ax + b)(cx + d)
\]

where \(a\), \(b\), \(c\), and \(d\) are constants.

### Step-by-Step Process:

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

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

   We need two numbers that multiply to 126 and add up to 27. The numbers \(9\) and \(14\) satisfy this condition because \(9 \times 14 = 126\) and \(9 + 14 = 27\).

3. **Rewrite the middle term using these factors:**
   \[
   9x^2 + 9x + 14x + 14
   \]

4. **Group and factor by grouping:**
   \[
   = 9x(x + 1) + 14(x + 1)
   \]

   Now, factor out the common term \((x + 1)\):

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

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

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

This trinomial is not prime, as it can be factored into two binomials.

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Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. How do you determine if a quadratic trinomial is factorable?
2. What are the steps for factoring a trinomial with a leading coefficient greater than 1?
3. Can all trinomials be factored using integers?
4. What is the significance of factoring trinomials in algebra?
5. How do you apply the distributive property in reverse to factor by grouping?

**Tip:** Always check your factored result by expanding it to see if it matches the original expression.

Learn how to factor the trinomial 9x^2 + 27x + 14 using step-by-step instructions. This problem covers key algebraic concepts, including factoring quadratic trinomials and grouping terms. Suitable for high school students studying algebra.

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