To subtract the given rational expressions and express it in the required form, we start with
\[ \frac{x+1}{x+3} - \frac{5}{2x+6}. \]
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Recognize the common denominator: The expression \( 2x+6 \) can be factored to \( 2(x+3) \).
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Rewrite both fractions with a common denominator: The common denominator between \( x+3 \) and \( 2(x+3) \) is \( 2(x+3) \).
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Adjust the first fraction: \[ \frac{x+1}{x+3} = \frac{x+1}{x+3} \cdot \frac{2}{2} = \frac{2(x+1)}{2(x+3)} = \frac{2x + 2}{2(x+3)}. \]
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Rewrite the second fraction with the common denominator: \[ \frac{5}{2x+6} = \frac{5}{2(x+3)}. \]
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Now subtract the fractions: \[ \frac{2x + 2}{2(x+3)} - \frac{5}{2(x+3)} = \frac{(2x + 2) - 5}{2(x+3)} = \frac{2x + 2 - 5}{2(x+3)} = \frac{2x - 3}{2(x+3)}. \]
Thus, the answer is:
\[ \frac{2x - 3}{2(x+3)}. \]
So, the missing part in your expression \( \frac{?}{2x+6} \) is \( 2x - 3 \). Therefore:
\[ \frac{2x - 3}{2(x+3)}. \]
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