Phương pháp giải:

$a{x^2} + bx + c < 0\forall x \in \mathbb{R}$$\Leftrightarrow \left\{ \begin{array}{l}a < 0\\\Delta ' < 0\end{array} \right.$

Lời giải chi tiết:

$\begin{array}{l}f\left( x \right) < 0\forall x \in \mathbb{R} \Leftrightarrow \left\{ \begin{array}{l}1 - m < 0\\\Delta ' < 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}m > 1\\{\left( {m - 1} \right)^2} + \left( {1 - m} \right) < 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}m > 1\\\left( {m - 1} \right)\left( {m - 2} \right) < 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}m > 1\\1 < m < 2\end{array} \right. \Leftrightarrow 1 < m < 2\end{array}$