Question

factor the following expression: \\(7x^3 + 28x^2 + 21x\\)

a \\(x(7x + 14x + 21)\\)

b \\(7(x^3 + 2x^2 + 3)\\)

c \\(7x(x^2 + 2x + 3)\\)

d \\(7x(x + 3)(x + 1)\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Factoring Trinomials",
"Quadratic Factoring"
],
"new_concepts": [
"Greatest Common Factor Factoring"
],
"current_concepts": [
"Greatest Common Factor Factoring",
"Factoring Trinomials",
"Quadratic Factoring"
]
}
</pre_analysis>

<reasoning>

Factor out the greatest common factor

\[
7x^3 + 28x^2 + 21x = 7x(x^2 + 4x + 3)
\]

Factor the remaining quadratic trinomial

\[
x^2 + 4x + 3 = (x + 3)(x + 1)
\]

Combine the factored components

\[
7x(x + 3)(x + 1)
\]
</reasoning>

<answer>
<mcq-option>(A) \(x(7x + 14x + 21)\)</mcq-option>
<mcq-option>(B) \(7(x^3 + 2x^2 + 3)\)</mcq-option>
<mcq-option>(C) \(7x(x^2 + 2x + 3)\)</mcq-option>
<mcq-correct>(D) \(7x(x + 3)(x + 1)\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Factoring"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Factoring Trinomials",
"Quadratic Factoring"
],
"new_concepts": [
"Greatest Common Factor Factoring"
],
"current_concepts": [
"Greatest Common Factor Factoring",
"Factoring Trinomials",
"Quadratic Factoring"
]
}
</pre_analysis>

<reasoning>

Factor out the greatest common factor

\[
7x^3 + 28x^2 + 21x = 7x(x^2 + 4x + 3)
\]

Factor the remaining quadratic trinomial

\[
x^2 + 4x + 3 = (x + 3)(x + 1)
\]

Combine the factored components

\[
7x(x + 3)(x + 1)
\]
</reasoning>

<answer>
<mcq-option>(A) \(x(7x + 14x + 21)\)</mcq-option>
<mcq-option>(B) \(7(x^3 + 2x^2 + 3)\)</mcq-option>
<mcq-option>(C) \(7x(x^2 + 2x + 3)\)</mcq-option>
<mcq-correct>(D) \(7x(x + 3)(x + 1)\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Factoring"
]
}
</post_analysis>