Question

current objective
simplify a rational expression with opposite factors

question

divide: \\(\frac{6x^2+5x-6}{-3x^2-10x+8}\\)

provide your answer below:

Explanation:

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"concepts_used": [
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"Simplifying Rational Expressions"
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</pre_analysis>

<reasoning>

Factor the numerator trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} 6x^2 + 5x - 6 &= 6x^2 + 9x - 4x - 6 \\ &= 3x(2x + 3) - 2(2x + 3) \\ &= (3x - 2)(2x + 3) \end{aligned}$$

\]

Factor the denominator trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} -3x^2 - 10x + 8 &= -(3x^2 + 10x - 8) \\ &= -(3x^2 + 12x - 2x - 8) \\ &= -[3x(x + 4) - 2(x + 4)] \\ &= -(3x - 2)(x + 4) \end{aligned}$$

\]

Simplify the rational expression

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{6x^2 + 5x - 6}{-3x^2 - 10x + 8} &= \frac{(3x - 2)(2x + 3)}{-(3x - 2)(x + 4)} \\ &= \frac{2x + 3}{-(x + 4)} \\ &= -\frac{2x + 3}{x + 4} \end{aligned}$$

\]
</reasoning>

<answer>
\(-\frac{2x + 3}{x + 4}\)
</answer>

<post_analysis>
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"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Simplifying Rational Expressions"
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</post_analysis>

Answer:

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"Opposite Factors"
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</pre_analysis>

<reasoning>

Factor the numerator trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} 6x^2 + 5x - 6 &= 6x^2 + 9x - 4x - 6 \\ &= 3x(2x + 3) - 2(2x + 3) \\ &= (3x - 2)(2x + 3) \end{aligned}$$

\]

Factor the denominator trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} -3x^2 - 10x + 8 &= -(3x^2 + 10x - 8) \\ &= -(3x^2 + 12x - 2x - 8) \\ &= -[3x(x + 4) - 2(x + 4)] \\ &= -(3x - 2)(x + 4) \end{aligned}$$

\]

Simplify the rational expression

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{6x^2 + 5x - 6}{-3x^2 - 10x + 8} &= \frac{(3x - 2)(2x + 3)}{-(3x - 2)(x + 4)} \\ &= \frac{2x + 3}{-(x + 4)} \\ &= -\frac{2x + 3}{x + 4} \end{aligned}$$

\]
</reasoning>

<answer>
\(-\frac{2x + 3}{x + 4}\)
</answer>

<post_analysis>
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"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Simplifying Rational Expressions"
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