$$\eqalign{
& \,\,\,\,3{x^2} + 4x + 10 \cr
& = 3\left( {{x^2} + {4 \over 3}x} \right) + 10 \cr
& = 3\left( {{x^2} + 2.x.{2 \over 3} + {4 \over 9} - {4 \over 9}} \right) + 10 \cr
& = 3{\left( {x + {2 \over 3}} \right)^2} - {4 \over 3} + 10 \cr
& = 3{\left( {x + {2 \over 3}} \right)^2} + {{26} \over 3} \cr
& Ta\,co:\,\,{\left( {x + {2 \over 3}} \right)^2} \ge 0\,\,\forall x \cr
& \Rightarrow 3{\left( {x + {2 \over 3}} \right)^2} \ge 0\,\,\forall x \Rightarrow {\left( {x + {2 \over 3}} \right)^2} + {{26} \over 3} \ge {{26} \over 3} > 0\,\,\forall x \cr
& Vay\,\,3{x^2} + 4x + 10 > 0\,\,\forall x. \cr} $$
