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Điều kiện: $7x^2+7>0.$
Khi đó ta có:
$(1) \Leftrightarrow \,\,\,\,7{x^2} + 7 \ge m{x^2} + 4x + m,\,\,\,\forall x \in \,R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$ \Leftrightarrow \left\{ \begin{array}{l}
\left( {7 - m} \right){x^2} - 4x + 7 - m \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\\
m{x^2} + 4x + m > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$\forall x \in \,R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$m = 7\,\,\,$$(2)$ không thỏa mãn $\forall x \in \,R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$m = 0$ $(3)$ không thỏa mãn $\forall x \in \,R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$(1)$ thỏa mãn $\forall x \in \,R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$ \Leftrightarrow \left\{ \begin{array}{l}
7 - m > 0\\
{\Delta _2}' = 4 - {\left( {7 - m} \right)^2} \le 0\\
m > 0\\
{\Delta _3}' = 4 - {m^2} < 0
\end{array} \right.$
$ \Leftrightarrow \left\{ \begin{array}{l}
m < 7\\
m \le 5\\
m > 0\\
m > 2
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, $
$\Leftrightarrow 2 < m \le 5$
Vậy giá trị cần tìm của m là $2 < m \le 5.$
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