Question

question

multiply: \\(\frac{y^2 - 4y - 45}{7y + 42} \cdot \frac{9y + 54}{-y^2 - 3y + 108}\\).

note: enter answer in factored form

provide your answer below:

Explanation:

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<reasoning>

Factor the first numerator

Using the Factoring Trinomials knowledge point
\[
y^2 - 4y - 45 = (y - 9)(y + 5)
\]

Factor the first denominator

Using the Greatest Common Factor Factoring knowledge point
\[
7y + 42 = 7(y + 6)
\]

Factor the second numerator

Using the Greatest Common Factor Factoring knowledge point
\[
9y + 54 = 9(y + 6)
\]

Factor the second denominator

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

$$\begin{aligned} -y^2 - 3y + 108 &= -(y^2 + 3y - 108) \\ &= -(y + 12)(y - 9) \end{aligned}$$

\]

Multiply and simplify the expressions

Using the Multiplying Rational Expressions and Simplifying Rational Expressions knowledge points
\[

$$\begin{aligned} \frac{y^2 - 4y - 45}{7y + 42} \cdot \frac{9y + 54}{-y^2 - 3y + 108} &= \frac{(y - 9)(y + 5)}{7(y + 6)} \cdot \frac{9(y + 6)}{-(y + 12)(y - 9)} \\ &= \frac{9(y - 9)(y + 5)(y + 6)}{-7(y + 6)(y + 12)(y - 9)} \\ &= -\frac{9(y + 5)}{7(y + 12)} \end{aligned}$$

\]
</reasoning>

<answer>
Multiply: \(\frac{y^2 - 4y - 45}{7y + 42} \cdot \frac{9y + 54}{-y^2 - 3y + 108}\).

Note: Enter answer in factored form

Provide your answer below:
<blank>\(-\frac{9(y + 5)}{7(y + 12)}\)</blank>
</answer>

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

Answer:

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<reasoning>

Factor the first numerator

Using the Factoring Trinomials knowledge point
\[
y^2 - 4y - 45 = (y - 9)(y + 5)
\]

Factor the first denominator

Using the Greatest Common Factor Factoring knowledge point
\[
7y + 42 = 7(y + 6)
\]

Factor the second numerator

Using the Greatest Common Factor Factoring knowledge point
\[
9y + 54 = 9(y + 6)
\]

Factor the second denominator

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

$$\begin{aligned} -y^2 - 3y + 108 &= -(y^2 + 3y - 108) \\ &= -(y + 12)(y - 9) \end{aligned}$$

\]

Multiply and simplify the expressions

Using the Multiplying Rational Expressions and Simplifying Rational Expressions knowledge points
\[

$$\begin{aligned} \frac{y^2 - 4y - 45}{7y + 42} \cdot \frac{9y + 54}{-y^2 - 3y + 108} &= \frac{(y - 9)(y + 5)}{7(y + 6)} \cdot \frac{9(y + 6)}{-(y + 12)(y - 9)} \\ &= \frac{9(y - 9)(y + 5)(y + 6)}{-7(y + 6)(y + 12)(y - 9)} \\ &= -\frac{9(y + 5)}{7(y + 12)} \end{aligned}$$

\]
</reasoning>

<answer>
Multiply: \(\frac{y^2 - 4y - 45}{7y + 42} \cdot \frac{9y + 54}{-y^2 - 3y + 108}\).

Note: Enter answer in factored form

Provide your answer below:
<blank>\(-\frac{9(y + 5)}{7(y + 12)}\)</blank>
</answer>

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"Algebra",
"Multiplying Rational Expressions"
]
}
</post_analysis>