Question

first up, lets review the assignm

get familiar with this topic by reviewing instruc

question

divide: \\(\frac{x^2-4x-32}{-x^2-7x-12}\\)

provide your answer below:

Explanation:

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<reasoning>

Factor the numerator trinomial

Using the Factoring Trinomials knowledge point
\[
x^2 - 4x - 32 = (x - 8)(x + 4)
\]

Factor the denominator trinomial

Using the Factoring Trinomials knowledge point
\[
-x^2 - 7x - 12 = -(x^2 + 7x + 12) = -(x + 3)(x + 4)
\]

Simplify the rational expression

Using the Simplifying Rational Expressions knowledge point
\[
\frac{x^2 - 4x - 32}{-x^2 - 7x - 12} = \frac{(x - 8)(x + 4)}{-(x + 3)(x + 4)} = \frac{x - 8}{-(x + 3)} = -\frac{x - 8}{x + 3} = \frac{8 - x}{x + 3}
\]
</reasoning>

<answer>
Divide: \(\frac{x^2-4x-32}{-x^2-7x-12}\)

Provide your answer below:
<blank>\(-\frac{x-8}{x+3}\)</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
"Algebra",
"Simplifying Rational Expressions"
]
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</post_analysis>

Answer:

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<reasoning>

Factor the numerator trinomial

Using the Factoring Trinomials knowledge point
\[
x^2 - 4x - 32 = (x - 8)(x + 4)
\]

Factor the denominator trinomial

Using the Factoring Trinomials knowledge point
\[
-x^2 - 7x - 12 = -(x^2 + 7x + 12) = -(x + 3)(x + 4)
\]

Simplify the rational expression

Using the Simplifying Rational Expressions knowledge point
\[
\frac{x^2 - 4x - 32}{-x^2 - 7x - 12} = \frac{(x - 8)(x + 4)}{-(x + 3)(x + 4)} = \frac{x - 8}{-(x + 3)} = -\frac{x - 8}{x + 3} = \frac{8 - x}{x + 3}
\]
</reasoning>

<answer>
Divide: \(\frac{x^2-4x-32}{-x^2-7x-12}\)

Provide your answer below:
<blank>\(-\frac{x-8}{x+3}\)</blank>
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Simplifying Rational Expressions"
]
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</post_analysis>