Question

Use the grouping method to factor the polynomial below completely.

x3 + 2x2 + 3x + 6
Group of answer choices

(x2 + 3)(x + 3)

(x2 + 2)(x + 3)

(x2 + 2)(x + 2)

(x2 + 3)(x + 2)

Answer 1

To factor the polynomial \(x^3 + 2x^2 + 3x + 6\) using the grouping method, we'll start by grouping the terms.

1. **Group the terms**:
\[
(x^3 + 2x^2) + (3x + 6)
\]

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

3. **Rewrite the polynomial with the factored groups**:
\[
x^2(x + 2) + 3(x + 2)
\]

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

Now, let's check the answer choices. The factored form we obtained is \((x + 2)(x^2 + 3)\), but this is not among the options you provided.

It appears that none of the answer choices perfectly match \((x + 2)(x^2 + 3)\). However, if we consider factoring just \(x^3 + 2x^2 + 3x + 6\) might not have the clean options given. If we only look for quadratics that can yield \(x^2+3\) or kind it might need, we also find that unless there had been a discounting of constants in solutions reviewed.

To summarize, the correct factorization of the given polynomial is:
\[
(x + 2)(x^2 + 3)
\]
Since it doesn't match any of the provided choices, it could be due to misinterpretation in the options or a missing option in provided answers. Double-checking your text or choices would be recommended.