Question

what is the difference?

\\(\frac{x+5}{x+2} - \frac{x+1}{x^2+2x}\\)

\\(\circ\\) \\(\frac{x^2+4x-1}{x(x+2)}\\)

\\(\circ\\) \\(\frac{x^2+4x+1}{x(x+2)}\\)

\\(\circ\\) \\(\frac{4}{-1(x^2+x-2)}\\)

\\(\circ\\) \\(\frac{x^2+6x+1}{x(x+2)}\\)

Explanation:

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<reasoning>

Find the least common denominator

\[

$$\begin{aligned} &\text{Denominators: } x+2 \text{ and } x^2+2x = x(x+2)\\ &\text{LCD: } x(x+2) \end{aligned}$$

\]

Rewrite fractions with the LCD

\[

$$\begin{aligned} &\frac{x+5}{x+2} \cdot \frac{x}{x} = \frac{x(x+5)}{x(x+2)} = \frac{x^2+5x}{x(x+2)}\\ &\frac{x+1}{x^2+2x} = \frac{x+1}{x(x+2)} \end{aligned}$$

\]

Subtract the numerators

\[

$$\begin{aligned} &\frac{x^2+5x}{x(x+2)} - \frac{x+1}{x(x+2)} = \frac{(x^2+5x) - (x+1)}{x(x+2)}\\ &\frac{x^2+5x-x-1}{x(x+2)} = \frac{x^2+4x-1}{x(x+2)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{x^2+4x-1}{x(x+2)}\)</mcq-correct>
<mcq-option>(B) \(\frac{x^2+4x+1}{x(x+2)}\)</mcq-option>
<mcq-option>(C) \(\frac{4}{-1(x^2+x-2)}\)</mcq-option>
<mcq-option>(D) \(\frac{x^2+6x+1}{x(x+2)}\)</mcq-option>
</answer>

<post_analysis>
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"Mathematics",
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"Rational Subtraction"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Find the least common denominator

\[

$$\begin{aligned} &\text{Denominators: } x+2 \text{ and } x^2+2x = x(x+2)\\ &\text{LCD: } x(x+2) \end{aligned}$$

\]

Rewrite fractions with the LCD

\[

$$\begin{aligned} &\frac{x+5}{x+2} \cdot \frac{x}{x} = \frac{x(x+5)}{x(x+2)} = \frac{x^2+5x}{x(x+2)}\\ &\frac{x+1}{x^2+2x} = \frac{x+1}{x(x+2)} \end{aligned}$$

\]

Subtract the numerators

\[

$$\begin{aligned} &\frac{x^2+5x}{x(x+2)} - \frac{x+1}{x(x+2)} = \frac{(x^2+5x) - (x+1)}{x(x+2)}\\ &\frac{x^2+5x-x-1}{x(x+2)} = \frac{x^2+4x-1}{x(x+2)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{x^2+4x-1}{x(x+2)}\)</mcq-correct>
<mcq-option>(B) \(\frac{x^2+4x+1}{x(x+2)}\)</mcq-option>
<mcq-option>(C) \(\frac{4}{-1(x^2+x-2)}\)</mcq-option>
<mcq-option>(D) \(\frac{x^2+6x+1}{x(x+2)}\)</mcq-option>
</answer>

<post_analysis>
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