a.
\(\Delta'=m^2-\left(3m-2\right)>0\Rightarrow\left[{}\begin{matrix}m>2\\m< 1\end{matrix}\right.\)
Theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=2m\\x_1x_2=3m-2\end{matrix}\right.\)
\(x_1^2+x_2^2+3x_1x_2=5\)
\(\Leftrightarrow\left(x_1+x_2\right)^2+x_1x_2=5\)
\(\Leftrightarrow4m^2+3m-2=5\)
\(\Leftrightarrow4m^2+3m-7=0\Rightarrow\left[{}\begin{matrix}m=1\left(loại\right)\\m=-\dfrac{7}{4}\end{matrix}\right.\)
b.
\(\Delta=\left(m+2\right)^2-8=m^2+4m-4>0\Rightarrow\left[{}\begin{matrix}m>-2+2\sqrt{2}\\m< -2-2\sqrt{2}\end{matrix}\right.\)
Theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=m+2\\x_1x_2=2\end{matrix}\right.\)
\(\dfrac{x_1}{x_2}+\dfrac{x_2}{x_1}=\dfrac{9}{2}\)
\(\Leftrightarrow x_1^2+x_2^2=\dfrac{9}{2}x_1x_2=9\)
\(\Leftrightarrow\left(x_1+x_2\right)^2-2x_1x_2=9\)
\(\Leftrightarrow\left(m+2\right)^2-4=9\)
\(\Leftrightarrow\left(m+2\right)^2=13\)
\(\Rightarrow\left[{}\begin{matrix}m=-2-\sqrt{13}\\m=-2+\sqrt{13}\end{matrix}\right.\)
c.
\(\Delta'=4-\left(m-1\right)>0\Rightarrow m< 5\)
Theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=4\\x_1x_2=m-1\end{matrix}\right.\)
\(x_1^3+x_2^3=40\)
\(\Leftrightarrow\left(x_1+x_2\right)^3-3x_1x_2\left(x_1+x_2\right)=40\)
\(\Leftrightarrow4^3-12\left(m-1\right)=40\)
\(\Rightarrow m=3\)
