\(m\ne-1\) ; \(\Delta'=\left(m-3\right)^2-9\left(m+1\right)=m^2-15m\)
a/ Để pt có 2 nghiệm dương pb
\(\Leftrightarrow\left\{{}\begin{matrix}\Delta'>0\\x_1+x_2>0\\x_1x_2>0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m^2-15m>0\\\frac{2\left(m-3\right)}{m+1}>0\\\frac{9}{m+1}>0\end{matrix}\right.\) \(m>15\)
b/ Để pt có 2 nghiệm pb thỏa \(x_1< -1< x_2\)
\(\Leftrightarrow f\left(-1\right)< 0\)
\(\Leftrightarrow\left(m+1\right).1+2\left(m-3\right)+9< 0\)
\(\Leftrightarrow3m+4< 0\Rightarrow m< -\frac{4}{3}\)
c/ Để pt có 2 nghiệm pb thỏa \(x_1< x_2< 2\)
\(\Leftrightarrow\left\{{}\begin{matrix}\Delta'>0\\\left(x_1-2\right)\left(x_2-2\right)>0\\\frac{x_1+x_2}{2}< 2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m^2-15m>0\\x_1x_2-2\left(x_1+x_2\right)+4>0\\x_1+x_2< 4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m^2-15m>0\\\frac{9}{m+1}+\frac{4\left(m-3\right)}{m+1}+4>0\\\frac{2\left(m-3\right)}{m+1}-4< 0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m^2-15m>0\\\frac{8m+1}{m+1}>0\\\frac{-2\left(m+5\right)}{m+1}< 0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}m< -5\\-1< m< -\frac{1}{8}\\m>15\end{matrix}\right.\)
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