Giải thích các bước giải:

Sửa đề: B4: Tìm m để $m{x^2} - 4\left( {m + 1} \right)x + m - 5 \le 0,\forall x \in R$

$\begin{array}{l}
B1:\\
a)\left( {m - 1} \right){x^2} - 2\left( {m + 1} \right)x - 3\left( {m - 2} \right) > 0,\forall x \in R\\
 \Leftrightarrow \left\{ \begin{array}{l}
m - 1 > 0\\
\Delta ' < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > 1\\
{\left( { - \left( {m + 1} \right)} \right)^2} - \left( {m - 1} \right).\left( { - 3\left( {m - 2} \right)} \right) > 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > 1\\
4{m^2} - 7m + 7 > 0\left( {ld} \right)
\end{array} \right.\\
 \Leftrightarrow m > 1\\
 \Leftrightarrow m \in \left( {1; + \infty } \right)\\
b)m{x^2} - 4\left( {m + 1} \right)x + m - 3 < 0,\forall x \in R\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
\Delta ' < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
{\left( { - 2\left( {m + 1} \right)} \right)^2} - m.\left( {m - 3} \right) < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
3{m^2} + 11m + 4 < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
\dfrac{{ - 11 - \sqrt {73} }}{6} < m < \dfrac{{ - 11 + \sqrt {73} }}{6}
\end{array} \right.\\
 \Leftrightarrow \dfrac{{ - 11 - \sqrt {73} }}{6} < m < \dfrac{{ - 11 + \sqrt {73} }}{6}\\
 \Leftrightarrow m \in \left( {\dfrac{{ - 11 - \sqrt {73} }}{6};\dfrac{{ - 11 + \sqrt {73} }}{6}} \right)
\end{array}$

$B2:$

$\left( {m - 3} \right){x^2} + \left( {m + 2} \right)x - 4 > 0$ vô nghiệm

$\begin{array}{l}
 \Leftrightarrow \left( {m - 3} \right){x^2} + \left( {m + 2} \right)x - 4 \le 0,\forall x \in R\\
 \Leftrightarrow \left\{ \begin{array}{l}
m - 3 < 0\\
\Delta  \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 3\\
{\left( {m + 2} \right)^2} - 4.\left( {m - 3} \right).\left( { - 4} \right) \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 3\\
{m^2} + 20m - 44 \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 3\\
 - 22 \le m \le 2
\end{array} \right.\\
 \Leftrightarrow  - 22 \le m \le 2\\
 \Leftrightarrow m \in \left[ { - 22;2} \right]
\end{array}$

$\begin{array}{l}
B3:\\
m{x^2} + 4x + m \ge 0,\forall x \in R\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > 0\\
\Delta ' \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > 0\\
{2^2} - m.m \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > 0\\
{m^2} \ge 4
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m > 0\\
\left[ \begin{array}{l}
m \ge 2\\
m \le  - 2
\end{array} \right.
\end{array} \right.\\
 \Leftrightarrow m \ge 2\\
 \Leftrightarrow m \in \left[ {2; + \infty } \right)
\end{array}$

$\begin{array}{l}
B4:\\
m{x^2} - 4\left( {m + 1} \right)x + m - 5 \le 0,\forall x \in R\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
\Delta ' \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
{\left( { - 2\left( {m + 1} \right)} \right)^2} - m.\left( {m - 5} \right) \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
3{m^2} + 13m + 4 \le 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
 - 4 \le m \le \dfrac{{ - 1}}{3}
\end{array} \right.\\
 \Leftrightarrow  - 4 \le m \le \dfrac{{ - 1}}{3}\\
 \Leftrightarrow m \in \left[ { - 4;\dfrac{{ - 1}}{3}} \right]
\end{array}$