Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
f\left( x \right) = a\,{x^2} + bx + c > 0,\,\,\,\forall x \in R \Leftrightarrow \left\{ \begin{array}{l}
a > 0\\
\Delta  < 0
\end{array} \right.\\
1.1,\\
f\left( x \right) > 0,\,\,\,\,\,\forall x \in R\\
 \Leftrightarrow {x^2} + 2\left( {m - 1} \right)x + m + 5 > 0,\,\,\,\,\,\forall x \in R\\
 \Leftrightarrow \Delta ' < 0\\
 \Leftrightarrow {\left( {m - 1} \right)^2} - 1.\left( {m + 5} \right) < 0\\
 \Leftrightarrow {m^2} - 2m + 1 - m - 5 < 0\\
 \Leftrightarrow {m^2} - 3m - 4 < 0\\
 \Leftrightarrow \left( {m - 4} \right)\left( {m + 1} \right) < 0\\
 \Leftrightarrow  - 1 < m < 4\\
*)\\
f\left( x \right) = a\,{x^2} + bx + c < 0,\,\,\,\forall x \in R \Leftrightarrow \left\{ \begin{array}{l}
a < 0\\
\Delta  < 0
\end{array} \right.\\
2.1\\
f\left( x \right) < 0,\,\,\,\,\forall x \in R\\
 \Leftrightarrow m{x^2} - 4\left( {m + 1} \right)x + m - 5 < 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 - 5} \right) < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
4{m^2} + 8m + 4 - {m^2} + 5m < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
3{m^2} + 13m + 4 < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
\left( {3m + 1} \right)\left( {m + 4} \right) < 0
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
m < 0\\
 - 4 < m <  - \frac{1}{3}
\end{array} \right.\\
 \Leftrightarrow  - 4 < m <  - \frac{1}{3}\\
2.2\\
f\left( x \right) < 0,\,\,\,\,\,\,\forall x \in R\\
 \Leftrightarrow  - {x^2} - 2\left( {m + 1} \right)x - 2m - 2 < 0,\,\,\,\,\,\,\,\,\forall x \in R\\
 \Leftrightarrow {x^2} + 2\left( {m + 1} \right)x + 2m + 2 > 0,\,\,\,\,\,\,\,\forall x \in R\\
 \Leftrightarrow \Delta ' < 0\\
 \Leftrightarrow {\left( {m + 1} \right)^2} - 1.\left( {2m + 1} \right) < 0\\
 \Leftrightarrow {m^2} + 2m + 1 - 2m - 2 < 0\\
 \Leftrightarrow {m^2} - 1 < 0\\
 \Leftrightarrow  - 1 < m < 1
\end{array}\)