$x.(x+1)<0$
$⇔ \left[ \begin{array}{l}\left\{ \begin{array}{l}x>0\\x+1<0\end{array} \right.\\\left\{ \begin{array}{l}x<0\\x+1>0\end{array} \right.\end{array} \right.$
$⇔ \left[ \begin{array}{l}\left\{ \begin{array}{l}x>0\\x<-1\end{array} \right.\\\left\{ \begin{array}{l}x<0\\x>-1\end{array} \right.\end{array} \right.$
$⇒ -1<x<0$