\(f\left(x\right)=\left(m-4\right)x^2+\left(m+1\right)x+2m-1\)

\(f\left(x\right)< 0,\forall x\in R\Leftrightarrow\left\{{}\begin{matrix}a< 0\\\Delta< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m-4< 0\\\left(m+1\right)^2-4\left(m-4\right)\left(2m-1\right)< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m< 4\\m^2+2m+1-4\left(2m^2-m-8m+4\right)< 0\end{matrix}\right.\)

\(\Leftrightarrow m^2+2m+1-8m^2+36m-16< 0\)

\(\Leftrightarrow-7m^2+38m-15< 0\)

\(\Leftrightarrow\left\{{}\begin{matrix}m< 4\\\left[{}\begin{matrix}m< \dfrac{3}{7}\\m>5\end{matrix}\right.\end{matrix}\right.\)

\(KL:m\in\left(5;+\infty\right)\)

\(a=-1< 0;\Delta=\left(2\sqrt{m}-1\right)^2+4\left(\sqrt{m}-m\right)=4m-4\sqrt{m}+1+4\sqrt{m}-4m=1>0\)

a/ \(f\left(x\right)\ge0\) vô nghiệm \(\Leftrightarrow f\left(x\right)< 0,\forall x\in R\Leftrightarrow\left\{{}\begin{matrix}a=-1< 0\left(tm\right)\\\Delta< 0\left(voly\right)\end{matrix}\right.\)

Vậy ko tồn tại m để ....

b/ \(f\left(x\right)\ge0,\forall x\in\left[1;2\right]\)

\(\Leftrightarrow\left\{{}\begin{matrix}\Delta>0\\\left[{}\begin{matrix}1< x_1< x_2\\x_1< x_2< 2\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}-1.f\left(1\right)>0\\\dfrac{x_1+x_2}{2}-1>0\end{matrix}\right.\\\left\{{}\begin{matrix}-1.f\left(2\right)>0\\\dfrac{x_1+x_2}{2}-2< 0\end{matrix}\right.\end{matrix}\right.\)

\(\left(1\right)\left\{{}\begin{matrix}-1+2\sqrt{m}-1-m+\sqrt{m}< 0\\\sqrt{m}-\dfrac{1}{2}-1>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m-3\sqrt{m}+2>0\\\sqrt{m}>\dfrac{3}{2}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}0< m< 1\\m>2\end{matrix}\right.\\m>\dfrac{9}{4}\end{matrix}\right.\Leftrightarrow m>\dfrac{9}{4}\)

\(\left(2\right)\left\{{}\begin{matrix}-4+4\sqrt{m}-2-m+\sqrt{m}< 0\\\sqrt{m}-\dfrac{1}{2}-2< 0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m-5\sqrt{m}+6>0\\\sqrt{m}< \dfrac{5}{2}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}0< m< 2\\m>3\end{matrix}\right.\\0\le m< \dfrac{25}{4}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}0< m< 2\\3< m< \dfrac{25}{4}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}m>\dfrac{9}{4}\\0< m< 2\\3< m< \dfrac{25}{4}\end{matrix}\right.\)

TH1: \(\Delta'=9+9m\le0\Rightarrow m\le-1\)

TH2: \(\left\{{}\begin{matrix}\Delta'>0\\-1\le x_1< x_2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m>-1\\\left(x_1+1\right)\left(x_2+1\right)\ge0\\\frac{x_1+x_2}{2}>-1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m>-1\\x_1x_2+x_1+x_2+1\ge0\\x_1+x_2>-2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m>-1\\-m+2+1\ge0\\2>-1\end{matrix}\right.\) \(\Rightarrow-1< m\le3\)

Vậy \(m\le3\)

Lời giải: 

Với $x>1$

$f(x)=m(x^2+2x+1)-2x+3>0\Leftrightarrow m>\frac{2x-3}{(x+1)^2}$

$\Leftrightarrow m>\frac{2x-3}{(x+1)^2}(\max)$ khi $x>1$

Xét $g(x)=\frac{2x-3}{(x+1)^2}$ với $x>1$

$g(x)=\frac{2(x+1)-5}{(x+1)^2}=\frac{2}{x+1}-\frac{5}{(x+1)^2}=\frac{1}{5}-5(\frac{1}{x+1}-\frac{1}{5})^2\leq \frac{1}{5}$ với mọi $x>1$

Do đó: $m>\frac{1}{5}$

\(f\left(x\right)>0,\forall x\in R\Leftrightarrow\left\{{}\begin{matrix}a>0\\\Delta< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m+1>0\\\left[-2\left(m-1\right)\right]^2-4\left(m+1\right)\left(-m+4\right)< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m>-1\\4\left(m^2-2m+1\right)-4\left(-m^2+4m-m+4\right)< 0\end{matrix}\right.\)

\(\Leftrightarrow4m^2-8m+4+4m^2-12m-16< 0\)

\(\Leftrightarrow8m^2-20m-12< 0\)

\(KL:m\in\left(-1;3\right)\)