To subtract the rational expressions \(\frac{x + 1}{x + 3} - \frac{5}{2x + 6}\), we first need to find a common denominator.
The second fraction can be simplified since \(2x + 6\) can be factored:
\[ 2x + 6 = 2(x + 3) \]
So the expression becomes:
\[ \frac{x + 1}{x + 3} - \frac{5}{2(x + 3)} \]
Now, we can see that the common denominator is \(2(x + 3)\). We will rewrite both fractions with this common denominator:
- The first fraction:
\[ \frac{x + 1}{x + 3} = \frac{(x + 1) \cdot 2}{2(x + 3)} = \frac{2(x + 1)}{2(x + 3)} = \frac{2x + 2}{2(x + 3)} \]
- The second fraction is already in the desired form:
\[ \frac{-5}{2(x + 3)} \]
Now we can combine the two fractions:
\[ \frac{2x + 2 - 5}{2(x + 3)} = \frac{2x + 2 - 5}{2(x + 3)} = \frac{2x - 3}{2(x + 3)} \]
Thus, the result of the subtraction is:
\[ \frac{2x - 3}{2(x + 3)} \]