$$\begin{gathered} \frac{x+3}{x+2}-\frac{x+4}{x+3}=\frac{x+5}{x+4}-\frac{x+6}{x+5} \\ \Rightarrow \frac{x+2+1}{x+2} + \frac{1}{x+2} - \frac{x+3}{x+3} - \frac{1}{x+3}= \frac{x+4+1}{x+4} + \frac{x+5+1}{x+5} \\ \Rightarrow \frac{x+2}{x+2} + \frac{1}{x+2} - \frac{x+3}{x+3} - \frac{1}{x+3} = \frac{x+4}{x+4} + \frac{1}{x+4} - \frac{x+5}{x+5} - \frac{1}{x+3} \\ \Rightarrow 1 + \frac{1}{x+2} -1 - \frac{1}{x+3} = 1 + \frac{1}{x+4} - 1 - \frac{1}{x+5} \\ \Rightarrow \frac{1}{x+2} - \frac{1}{x+3} = \frac{1}{x+4} - \frac{1}{x+5} \\ \Rightarrow \frac{x+3 - \left(x+2\right)}{\left(x+2\right) \left(x+3\right)} = \frac{x+5 -\left(x+4\right) }{\left(x+4\right) \left(x+5\right)} \\ \Rightarrow \frac{x+3 - x-2}{\left(x+2\right) \left(x+3\right)} = \frac{x+5 -x-4}{\left(x+4\right) \left(x+5\right)} \\ \Rightarrow \frac{1}{\left(x+2\right) \left(x+3\right)} = \frac{1}{\left(x+4\right) \left(x+5\right)} \\ \Rightarrow \left(x+4\right) \left(x+5\right) = \left(x+2\right) \left(x+3\right) \\ \Rightarrow x^2 + 5x + 4x + 20 = x^2+3x+2x+6 \\ \Rightarrow x^2 + 9x - x^2 -5x = 6-20 \\ \Rightarrow 4x = -14 \\ \therefore x = \frac{-14}{4} = \frac{-7}{2} \end{gathered}$$