Đáp án:
$\begin{array}{l}
1)\\
\left( {{x^2} + 2x + 3} \right)\left( {{x^2} + 2x + 4} \right) + 3\\
Do:\left\{ \begin{array}{l}
{x^2} + 2x + 3 = {x^2} + 2x + 1 + 2\\
= {\left( {x + 1} \right)^2} + 2 \ge 2 > 0\\
{x^2} + 2x + 4 = {\left( {x + 1} \right)^2} + 3 > 0
\end{array} \right.\\
\Rightarrow \left( {{x^2} + 2x + 3} \right)\left( {{x^2} + 2x + 4} \right) > 0\forall x\\
\Rightarrow \left( {{x^2} + 2x + 3} \right)\left( {{x^2} + 2x + 4} \right) + 3 > 0\forall x\\
2)\\
{\left( {a + b + c} \right)^2} + {\left( {a - b - c} \right)^2}\\
+ {\left( {b - c - a} \right)^2} + {\left( {c - a - b} \right)^2}\\
= {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac\\
+ {a^2} + {b^2} + {c^2} - 2ab - 2ac + 2bc\\
+ {b^2} + {c^2} + {a^2} - 2ab - 2bc + 2ac\\
+ {c^2} + {a^2} + {b^2} - 2ac - 2bc + 2ab\\
= 4\left( {{a^2} + {b^2} + {c^2}} \right)
\end{array}$
