1/$f(x)=(m-1)x^2-2(m+1)x+3(m-2)$ luôn dương với mọi $x$ khi

$\left\{\begin{matrix}m-1>0\\(2m+2)^2-4(m-1)3(m-2)<0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m>1\\4m^2+8m+4-12m^2+36m-24<0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m>1\\-8m^2+44m-20<0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m>1\\\left[\begin{matrix}m<\dfrac{1}{2}\\m>5\end{matrix}\right.\end{matrix}\right.$

$\Leftrightarrow m>5$

2/$f(x)=(m-4)x^2+(m+1)x+2m-1$ luôn âm với mọi $x$ khi

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

$\Leftrightarrow \left\{\begin{matrix}m<4\\m^2+2m+1-8m^2+36m-16<0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m<4\\-7m^2+38m-15<0\end{matrix}\right.$

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

$\Leftrightarrow m<\dfrac{3}{7}$

3/a)$\left\{\begin{matrix}m+1>0\\(2m-2)^2-4(m+1)(3m-3)\le0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m>-1\\4m^2-8m+4-12m^2+12\le0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m>-1\\-8m^2-8m+16\le0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m>-1\\\left[\begin{matrix}m\le-2\\m\ge1\end{matrix}\right.\end{matrix}\right.$

$\Leftrightarrow m\ge1$

b)$\left[\begin{matrix}\left\{\begin{matrix}x^2-8x+20<0\\mx^2+2(m+1)x+9m+4>0\end{matrix}\right.\\\left\{\begin{matrix}x^2-8x+20>0\\mx^2+2(m+1)x+9m+4<0\end{matrix}\right.\end{matrix}\right.$

Vì $x^2-8x+20>0\ \forall x\in R$ nên để bpt nghiệm đúng với mọi $x$ thì $mx^2+2(m+1)x+9m+4<0\ \forall x\in R$

$\left\{\begin{matrix}m<0\\(2m+2)^2-4m(9m+4)<0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m<0\\4m^2+8m+4-36m^2-16m<0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m<0\\-32m^2-8m+4<0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m<0\\\left[\begin{matrix}m<\dfrac{-1}{2}\\m>\dfrac{1}{4}\end{matrix}\right.\end{matrix}\right.$

$\Leftrightarrow m<\dfrac{-1}{2}$

4/a)$\left\{\begin{matrix}9m-5>0\\2(m+1)<0\\4m^2+8m+4-36m+20>0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m>\dfrac{5}{9}\\m<-1\\\left[\begin{matrix}m<3-\sqrt{3}\\m>3+\sqrt{3}\end{matrix}\right.\end{matrix}\right.$

$\Rightarrow$ không có $m$ thỏa mãn

b)$\left\{\begin{matrix}4m^2-4m^2-4m+24>0\\\dfrac{2m}{m-2}>0\\\dfrac{m+3}{m-2}>0\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}m<5\\\left[\begin{matrix}m>2\\m<0\end{matrix}\right.\\\left[\begin{matrix}m >2\\m<-3\end{matrix}\right.\end{matrix}\right.$

$\Leftrightarrow 2<m<5$

5/$\left\{\begin{matrix}x^2+10x+16\le0\\mx\ge3m+1\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}(x+2)(x+8)\le0\\x\ge\dfrac{3m+1}{m}\end{matrix}\right.$

$\Leftrightarrow \left\{\begin{matrix}-8\le x\le-2\\x\ge\dfrac{3m+1}{m}\end{matrix}\right.$

Để bpt vô nghiệm thì $\dfrac{3m+1}{m}>-2$

$\Leftrightarrow 3m+1>-2m\\\Leftrightarrow m>\dfrac{-1}{5}$