Question

current objective
divide rational expressions

question

divide and simplify:
\\\frac{3x^2 + 15x + 18}{x^2 + 2x - 24} \div \frac{x + 3}{x - 4}\\

Explanation:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{3x^2 + 15x + 18}{x^2 + 2x - 24} \div \frac{x + 3}{x - 4} = \frac{3x^2 + 15x + 18}{x^2 + 2x - 24} \cdot \frac{x - 4}{x + 3}
\]

Factor the first numerator

Using the Greatest Common Factor Factoring and Factoring Trinomials knowledge points
\[

$$\begin{aligned} 3x^2 + 15x + 18 &= 3(x^2 + 5x + 6) \\ &= 3(x + 2)(x + 3) \end{aligned}$$

\]

Factor the first denominator

Using the Factoring Trinomials knowledge point
\[
x^2 + 2x - 24 = (x + 6)(x - 4)
\]

Substitute and simplify

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{3(x + 2)(x + 3)}{(x + 6)(x - 4)} \cdot \frac{x - 4}{x + 3} &= \frac{3(x + 2)\cancel{(x + 3)}\cancel{(x - 4)}}{(x + 6)\cancel{(x - 4)}\cancel{(x + 3)}} \\ &= \frac{3(x + 2)}{x + 6} \end{aligned}$$

\]
</reasoning>

<answer>
\[\frac{3(x + 2)}{x + 6}\]
</answer>

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Answer:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{3x^2 + 15x + 18}{x^2 + 2x - 24} \div \frac{x + 3}{x - 4} = \frac{3x^2 + 15x + 18}{x^2 + 2x - 24} \cdot \frac{x - 4}{x + 3}
\]

Factor the first numerator

Using the Greatest Common Factor Factoring and Factoring Trinomials knowledge points
\[

$$\begin{aligned} 3x^2 + 15x + 18 &= 3(x^2 + 5x + 6) \\ &= 3(x + 2)(x + 3) \end{aligned}$$

\]

Factor the first denominator

Using the Factoring Trinomials knowledge point
\[
x^2 + 2x - 24 = (x + 6)(x - 4)
\]

Substitute and simplify

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{3(x + 2)(x + 3)}{(x + 6)(x - 4)} \cdot \frac{x - 4}{x + 3} &= \frac{3(x + 2)\cancel{(x + 3)}\cancel{(x - 4)}}{(x + 6)\cancel{(x - 4)}\cancel{(x + 3)}} \\ &= \frac{3(x + 2)}{x + 6} \end{aligned}$$

\]
</reasoning>

<answer>
\[\frac{3(x + 2)}{x + 6}\]
</answer>

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