Chọn D
TXĐ: \(D={\rm R}\backslash \left\{-m\right\}\)
\(y=\frac{\left(m+1\right)x+2m+2}{x+m} \Rightarrow y'=\frac{\left(m+1\right)m-2m-2}{\left(x+m\right)^{2} } =\frac{m^{2} -m-2}{\left(x+m\right)^{2} } \).
Hàm số nghịch biến trên \(\left(-1;+\infty \right)\Leftrightarrow y'<0,\forall x\in \left(-1;+\infty \right)\)
\(\Leftrightarrow \left\{\begin{array}{l} {-m\le -1} \\ {m^{2} -m-2<0} \end{array}\right. \Leftrightarrow \left\{\begin{array}{l} {m\ge 1} \\ {-1\le m<2} \end{array}\right. \Leftrightarrow 1\le m<2. \)