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The trinomial given is:

\[
3x^2 + 13x + 14
\]

To factor this, we use the method of splitting the middle term. We need two numbers that multiply to give \(3 \times 14 = 42\), and add to give \(13\).

The factors of 42 that add up to 13 are 6 and 7, since:

\[
6 + 7 = 13 \quad \text{and} \quad 6 \times 7 = 42
\]

Now, we rewrite the middle term \(13x\) as \(6x + 7x\):

\[
3x^2 + 6x + 7x + 14
\]

Next, we factor by grouping:

\[
(3x^2 + 6x) + (7x + 14)
\]

Factor out the greatest common factor from each group:

\[
3x(x + 2) + 7(x + 2)
\]

Now, factor out the common binomial factor \((x + 2)\):

\[
(3x + 7)(x + 2)
\]

Thus, the factored form of the trinomial is:

\[
(3x + 7)(x + 2)
\]

Would you like more details or further explanations?

### Related questions:
1. How do you verify that the factorization is correct?
2. Can all quadratic trinomials be factored by grouping?
3. How do you factor trinomials where the leading coefficient is 1?
4. What if no factors add up to the middle term in a trinomial? How do you proceed?
5. How is the discriminant used in factoring or solving quadratic equations?

### Tip:
Always check your factorization by expanding the factors back into their original form to confirm correctness.

Learn how to factor the trinomial 3x^2 + 13x + 14 with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring by grouping. Suitable for students in Grades 9-11.

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