https://file.aiquickdraw.com/mathbot/file/17260826585102fc6cpju.jpg

The problem displayed in the image is asking to factor the trinomial:

\[
3x^2 + 23x + 14
\]

To factor this trinomial, we need to find two numbers that multiply to give the product of the leading coefficient (3) and the constant term (14), which is \(3 \times 14 = 42\), and at the same time, add up to give the middle coefficient (23).

The two numbers that meet these conditions are 21 and 2 because:

\[
21 \times 2 = 42 \quad \text{and} \quad 21 + 2 = 23
\]

Now, we can rewrite the middle term \(23x\) as \(21x + 2x\):

\[
3x^2 + 21x + 2x + 14
\]

Next, group the terms in pairs:

\[
(3x^2 + 21x) + (2x + 14)
\]

Factor out the greatest common factor (GCF) from each group:

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

Now, factor out the common binomial factor \((x + 7)\):

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

Thus, the factored form of the trinomial is:

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

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How do you factor trinomials with negative coefficients?
2. What is the difference between factoring and solving a quadratic equation?
3. How do you factor a perfect square trinomial?
4. Can all trinomials be factored? If not, how do you determine that?
5. What is the process for factoring trinomials when the leading coefficient is negative?

**Tip**: Always check your factored answer by expanding it back out to ensure you get the original expression!

This lesson demonstrates how to factor the quadratic trinomial 3x^2 + 23x + 14 using the method of grouping. Learn step-by-step how to rewrite the middle term and apply common factoring.

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