Question

Use the grouping method to factor the polynomial below completely.



3x3 + 6x2 + 5x + 10

(3x2 + 2)(x + 2)

(3x2 + 5)(x + 5)

(3x2 + 2)(x + 5)

(3x2 + 5)(x + 2)

Answer 1

To factor the polynomial \( 3x^3 + 6x^2 + 5x + 10 \) using the grouping method, we follow these steps:

1. **Group the terms**: We can separate the polynomial into two groups:
\[
(3x^3 + 6x^2) + (5x + 10)
\]

2. **Factor out the common factors in each group**:
- For the first group \( 3x^3 + 6x^2 \), we can factor out \( 3x^2 \):
\[
3x^2(x + 2)
\]
- For the second group \( 5x + 10 \), we can factor out \( 5 \):
\[
5(x + 2)
\]

3. **Combine the factored groups**:
Now we have:
\[
3x^2(x + 2) + 5(x + 2)
\]

4. **Factor out the common binomial factor \( (x + 2) \)**:
\[
(x + 2)(3x^2 + 5)
\]

Thus, the factorization of the polynomial \( 3x^3 + 6x^2 + 5x + 10 \) is:
\[
(x + 2)(3x^2 + 5)
\]

From the provided options, none of them match this exact factorization. However, it seems that you are given choices that only present quadratic terms.

Let’s analyze our final expression against the options provided:
- \( (3x^2 + 2)(x + 2) \)
- \( (3x^2 + 5)(x + 5) \)
- \( (3x^2 + 2)(x + 5) \)
- \( (3x^2 + 5)(x + 2) \)

The equivalent factorization in terms of the valid choices from the set is \( (3x^2 + 5)(x + 2) \), which corresponds to our factored form:
\[
\text{Correct Answer: } (3x^2 + 5)(x + 2)
\]
This aligns with the recognition that \( (x + 2)(3x^2 + 5) \) factors identically.