a.
\(x^2+4\left(m+1\right)x+m^2+m>0;\forall x\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=1>0\\\Delta'=4\left(m+1\right)^2-\left(m^2+m\right)< 0\end{matrix}\right.\)
\(\Leftrightarrow\left(m+1\right)\left(3m+4\right)< 0\)
\(\Rightarrow-\dfrac{4}{3}< m< -1\)
b.
Đặt \(f\left(x\right)=3x^2-2\left(m+5\right)x-m^2+2m+8\)
BPT đúng với mọi \(x\in\left[-1;1\right]\) khi và chỉ khi:
\(\left\{{}\begin{matrix}f\left(-1\right)\le0\\f\left(1\right)\le0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}3-2\left(m+5\right)-m^2+2m+8\le0\\3+2\left(m+5\right)-m^2+2m+8\le0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m^2\ge1\\-m^2+4m+21\le0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}m\le-3\\m\ge7\end{matrix}\right.\)
