Đáp án:
$\begin{array}{l}
a)\Delta ' = {\left( {m - 1} \right)^2} + 2{m^2} - m + 1\\
= {m^2} - 2m + 1 + 2{m^2} - m + 1\\
= 3{m^2} - 3m + 2\\
= 3.\left( {{m^2} - m + \dfrac{2}{3}} \right)\\
= 3.\left[ {{{\left( {m - \dfrac{1}{2}} \right)}^2} - \dfrac{1}{4} + \dfrac{2}{3}} \right]\\
= 3.{\left( {m - \dfrac{1}{2}} \right)^2} + \dfrac{5}{4} > 0\forall m
\end{array}$
Vậy pt có 2 nghiệm phân biệt với mọi m
b)
$\begin{array}{l}
Theo\,Viet:\left\{ \begin{array}{l}
{x_1} + {x_2} = 2\left( {m - 1} \right)\\
{x_1}{x_2} = - 2{m^2} + m - 1
\end{array} \right.\\
A = x_1^2 + x_2^2 - 3{x_1}{x_2}\\
= {\left( {{x_1} + {x_2}} \right)^2} - 5{x_1}{x_2}\\
= 4{\left( {m - 1} \right)^2} - 5.\left( { - 2{m^2} + m - 1} \right)\\
= 4{m^2} - 8m + 4 + 10{m^2} - 5m + 5\\
= 14{m^2} - 13m + 9\\
= 14.\left( {{m^2} - 2.m.\dfrac{{13}}{{28}} + \dfrac{{{{13}^2}}}{{{{28}^2}}}} \right) + \dfrac{{335}}{{56}}\\
= 14.{\left( {m - \dfrac{{13}}{{28}}} \right)^2} + \dfrac{{335}}{{56}} \ge \dfrac{{335}}{{56}}\\
\Rightarrow \min A = \dfrac{{335}}{{56}}\\
c)B = - x_1^2 - x_2^2 + 4{x_1}{x_2}\\
= - {\left( {{x_1} + {x_2}} \right)^2} + 6{x_1}{x_2}\\
= - 4{\left( {m - 1} \right)^2} + 6.\left( { - 2{m^2} + m - 1} \right)\\
= - 4{m^2} + 8m - 4 - 12{m^2} + 6m - 6\\
= - 16{m^2} + 14m - 10\\
= - 16.\left( {{m^2} - \dfrac{7}{8}m + \dfrac{{{7^2}}}{{{{16}^2}}}} \right) - \dfrac{{111}}{{16}}\\
= - 16.{\left( {m - \dfrac{7}{{16}}} \right)^2} - \dfrac{{111}}{{16}} \le - \dfrac{{111}}{{16}}\\
\Rightarrow \max B = - \dfrac{{111}}{{16}}
\end{array}$
