To subtract the given rational expressions \(\frac{x+1}{x+3} - \frac{5}{2x+6}\), we first need to find a common denominator.
The expression \(2x + 6\) can be factored as:
\[ 2x + 6 = 2(x + 3) \]
So, the common denominator between \(x + 3\) and \(2(x + 3)\) is \(2(x + 3)\).
Now, we can rewrite both fractions with the common denominator:
- The first fraction \(\frac{x + 1}{x + 3}\) can be rewritten as:
\[ \frac{x + 1}{x + 3} = \frac{(x + 1) \cdot 2}{(x + 3) \cdot 2} = \frac{2(x + 1)}{2(x + 3)} \]
- The second fraction is already in the new common denominator:
\[ \frac{5}{2x + 6} = \frac{5}{2(x + 3)} \]
Now we can perform the subtraction:
\[ \frac{2(x + 1)}{2(x + 3)} - \frac{5}{2(x + 3)} = \frac{2(x + 1) - 5}{2(x + 3)} \]
Next, we simplify the numerator:
\[ 2(x + 1) - 5 = 2x + 2 - 5 = 2x - 3 \]
Therefore, we have:
\[ \frac{2(x + 1) - 5}{2(x + 3)} = \frac{2x - 3}{2(x + 3)} \]
Thus, the subtraction results in:
\[ \frac{2x - 3}{2x + 6} \]
So, the final answer is:
\[ \frac{2x - 3}{2x + 6} \]
This means that the box \(□\) can be filled with \(2x - 3\):
\[ \boxed{2x - 3} \]
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