a/ \(\left\{{}\begin{matrix}3m+1>0\\\Delta=\left(3m+1\right)^2-4\left(3m+1\right)\left(m+4\right)\le0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m>-\frac{1}{3}\\\left(3m+1\right)\left(-m-15\right)\le0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m>-\frac{1}{3}\\\left[{}\begin{matrix}m\ge-\frac{1}{3}\\m\le-15\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow m>-\frac{1}{3}\)
b/\(\left\{{}\begin{matrix}m+1>0\\\Delta'=\left(m-1\right)^2-\left(m+1\right)\left(3m-3\right)\le0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m>-1\\\left(m-1\right)\left(-2m-4\right)\le0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}m>-1\\\left[{}\begin{matrix}m\ge1\\m\le-2\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow m\ge1\)
