Question

current objective
simplify a rational expression with opposite factors

question
divide: \\(\frac{14m^2 - 11m - 60}{-12m^2 + 24m + 15}\\).

provide your answer below:

Explanation:

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</pre_analysis>

<reasoning>

Factor the numerator trinomial

Using the Trinomial Factorization knowledge point
\[

$$\begin{aligned} 14m^2 - 11m - 60 &= 14m^2 - 35m + 24m - 60 \\ &= 7m(2m - 5) + 12(2m - 5) \\ &= (2m - 5)(7m + 12) \end{aligned}$$

\]

Factor the denominator trinomial

Using the Greatest Common Factor Factoring and Trinomial Factorization knowledge points
\[

$$\begin{aligned} -12m^2 + 24m + 15 &= -3(4m^2 - 8m - 5) \\ &= -3(4m^2 - 10m + 2m - 5) \\ &= -3[2m(2m - 5) + 1(2m - 5)] \\ &= -3(2m - 5)(2m + 1) \end{aligned}$$

\]

Simplify the rational expression

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{14m^2 - 11m - 60}{-12m^2 + 24m + 15} &= \frac{(2m - 5)(7m + 12)}{-3(2m - 5)(2m + 1)} \\ &= \frac{7m + 12}{-3(2m + 1)} \\ &= -\frac{7m + 12}{3(2m + 1)} \end{aligned}$$

\]
</reasoning>

<answer>
\(-\frac{7m + 12}{3(2m + 1)}\)
</answer>

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Answer:

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<reasoning>

Factor the numerator trinomial

Using the Trinomial Factorization knowledge point
\[

$$\begin{aligned} 14m^2 - 11m - 60 &= 14m^2 - 35m + 24m - 60 \\ &= 7m(2m - 5) + 12(2m - 5) \\ &= (2m - 5)(7m + 12) \end{aligned}$$

\]

Factor the denominator trinomial

Using the Greatest Common Factor Factoring and Trinomial Factorization knowledge points
\[

$$\begin{aligned} -12m^2 + 24m + 15 &= -3(4m^2 - 8m - 5) \\ &= -3(4m^2 - 10m + 2m - 5) \\ &= -3[2m(2m - 5) + 1(2m - 5)] \\ &= -3(2m - 5)(2m + 1) \end{aligned}$$

\]

Simplify the rational expression

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{14m^2 - 11m - 60}{-12m^2 + 24m + 15} &= \frac{(2m - 5)(7m + 12)}{-3(2m - 5)(2m + 1)} \\ &= \frac{7m + 12}{-3(2m + 1)} \\ &= -\frac{7m + 12}{3(2m + 1)} \end{aligned}$$

\]
</reasoning>

<answer>
\(-\frac{7m + 12}{3(2m + 1)}\)
</answer>

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