Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
f\left( x \right) = - {x^2} + 2\left( {m + 1} \right)x - \left( {2{m^2} + m + 3} \right)\\
= - \left[ {{x^2} - 2\left( {m + 1} \right)x + \left( {2{m^2} + m + 3} \right)} \right]\\
= - \left[ {\left( {{x^2} - 2\left( {m + 1} \right)x + \left( {{m^2} + 2m + 1} \right)} \right) + \left( {{m^2} - m + 2} \right)} \right]\\
= - \left[ {{{\left( {x - m - 1} \right)}^2} + {{\left( {m - \frac{1}{2}} \right)}^2} + \frac{7}{4}} \right]\\
{\left( {x - m - 1} \right)^2} \ge 0,\,\,\,\,\forall x,m\\
{\left( {m - \frac{1}{2}} \right)^2} \ge 0,\,\,\,\,\,\forall m\\
\Rightarrow {\left( {x - m - 1} \right)^2} + {\left( {m - \frac{1}{2}} \right)^2} + \frac{7}{4} > 0,\,\,\,\,\forall x,m\\
\Rightarrow f\left( x \right) < 0,\,\,\,\,\,\forall x,m\\
\end{array}\)
