Đáp án:
$\begin{array}{l}
a)\forall m\\
b)\left\{ \begin{array}{l}
{x_1} + {x_2} = 2m + 4\\
{x_1}{x_2} = - 3{m^2} + 4m - 6
\end{array} \right.\\
S = 2\left( {x_1^2 + x_2^2} \right) - 3{x_1}{x_2}\\
= 2{\left( {{x_1} + {x_2}} \right)^2} - 2.2{x_1}{x_2} - 3{x_1}{x_2}\\
= 2.{\left( {2m + 4} \right)^2} - 5.\left( { - 3{m^2} + 4m - 6} \right)\\
= 8{m^2} + 32m + 32 + 15{m^2} - 20m + 30\\
= 23{m^2} + 12m + 62\\
= {\left( {\sqrt {23} m} \right)^2} + 2.\sqrt {23} m.\frac{6}{{\sqrt {23} }} + \frac{{36}}{{23}} + \frac{{1390}}{{23}}\\
= {\left( {\sqrt {23} m + \frac{6}{{\sqrt {23} }}} \right)^2} + \frac{{1390}}{{23}} \ge \frac{{1390}}{{23}}\\
\Rightarrow GTNN:S = \frac{{1390}}{{23}} \Leftrightarrow m = \frac{6}{{23}}
\end{array}$
