Same Bottom

Fractions with the same bottom add the way number fractions do: $\frac{2}{7}+\frac{3}{7} = \frac{5}{7}$. Keep the bottom and add the tops. The same holds with x in them:

$$\frac{x+5}{x+1}+\frac{2x-3}{x+1} = \frac{x+5+2x-3}{x+1}$$

Then collect like terms in the top: $\frac{3x+2}{x+1}$. Why keep the bottom? It names the size of the pieces. Both fractions count pieces of the same size, so adding counts pieces, and the size stays as it was. Adding the bottoms would give $\frac{2}{7}+\frac{3}{7}$ as 5/14, which is smaller than 3/7 alone.

Last, see whether the new top shares a factor with the bottom. $\frac{x+1}{x+2}+\frac{x+3}{x+2}$ gives $\frac{2x+4}{x+2}$, and the top is $2\left(x+2\right)$, so the whole sum cancels to 2.