Question

find all numbers that must be excluded from the domain of the rational expression.

\\\frac{x - 5}{x^2 + 5x + 4}\\

type the values for which the rational expression is undefined. select the correct choice below and fill in any answer boxes within your choice.

a. (use a comma to separate answers as needed.)
b. the rational expression is defined for all real numbers.

Explanation:

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<reasoning>

Identify the condition for domain exclusion

Using the Domain Exclusion knowledge point
\[
x^2 + 5x + 4 = 0
\]

Factor the quadratic denominator

Using the Factoring Quadratic Trinomials knowledge point
\[
(x + 4)(x + 1) = 0
\]

Solve for the excluded values

Using the Zero-Product Principle knowledge point
\[

$$\begin{aligned} x + 4 = 0 \quad &\Rightarrow \quad x = -4 \\ x + 1 = 0 \quad &\Rightarrow \quad x = -1 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(-4, -1\) (Use a comma to separate answers as needed.)</mcq-correct>
<mcq-option>(B) The rational expression is defined for all real numbers.</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the condition for domain exclusion

Using the Domain Exclusion knowledge point
\[
x^2 + 5x + 4 = 0
\]

Factor the quadratic denominator

Using the Factoring Quadratic Trinomials knowledge point
\[
(x + 4)(x + 1) = 0
\]

Solve for the excluded values

Using the Zero-Product Principle knowledge point
\[

$$\begin{aligned} x + 4 = 0 \quad &\Rightarrow \quad x = -4 \\ x + 1 = 0 \quad &\Rightarrow \quad x = -1 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(-4, -1\) (Use a comma to separate answers as needed.)</mcq-correct>
<mcq-option>(B) The rational expression is defined for all real numbers.</mcq-option>
</answer>

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