Đáp án:
\[ - 3 - 2\sqrt 2 \le m < - 9 + 2\sqrt {14} \]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
{x^2} - 2x + 3 = {\left( {x - 1} \right)^2} + 2 > 0,\,\,\,\forall x\\
1 \le \frac{{2{x^2} + \left( {m + 1} \right)x + 5}}{{{x^2} - 2x + 3}} < 4,\,\,\,\,\forall x \in R\\
\Leftrightarrow {x^2} - 2x + 3 \le 2{x^2} + \left( {m + 1} \right)x + 5 < 4{x^2} - 8x + 12,\,\,\,\,\forall x \in R\\
\Leftrightarrow \left\{ \begin{array}{l}
{x^2} + \left( {m + 3} \right)x + 2 \ge 0\,\,\,\,\,\,\left( 1 \right)\\
2{x^2} - \left( {m + 9} \right)x + 7 > 0\,\,\,\,\,\,\,\left( 2 \right)
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall x \in R\\
\left( 1 \right) Δ\Leftrightarrow \le 0\\
\Leftrightarrow {\left( {m + 3} \right)^2} - 4.2 \le 0\\
\Leftrightarrow {m^2} + 6m + 1 \le 0\\
\Leftrightarrow - 3 - 2\sqrt 2 \le m \le - 3 + 2\sqrt 2 \,\,\,\,\,\,\left( * \right)\\
\left( 2 \right)Δ \Leftrightarrow < 0\\
\Leftrightarrow {\left( {m + 9} \right)^2} - 4.2.7 < 0\\
\Leftrightarrow {m^2} + 18m + 81 - 56 < 0\\
\Leftrightarrow {m^2} + 18m + 25 < 0\\
\Leftrightarrow - 9 - 2\sqrt {14} < m < - 9 + 2\sqrt {14} \,\,\,\,\,\left( {**} \right)\\
\left( * \right);\left( {**} \right) \Rightarrow - 3 - 2\sqrt 2 \le m < - 9 + 2\sqrt {14}
\end{array}\)
