$\begin{array}{l} a){x^2} + x + 1 = {\left( {x + \frac{1}{2}} \right)^2} + \frac{3}{4} > 0,\forall x\\ b)\left( {x - 3} \right)\left( {x - 5} \right) + 2 = {x^2} - 8x + 17 = {\left( {x - 4} \right)^2} + 1 > 0,\forall x\\ c){x^2} + {y^2} + 2xy + 4 = {\left( {x + y} \right)^2} + 4 > 0,\forall x,y\\ d)4x - 10 - {x^2} = - \left( {{x^2} - 4x + 10} \right) = - {\left( {x - 2} \right)^2} - 6 < 0,\forall x\\ e) - {x^2} + 4x - 5 = - \left( {{x^2} - 4x + 5} \right) = - {\left( {x - 2} \right)^2} - 1 < 0,\forall x\\ f){x^2} + 2x + {y^2} + 1 = {\left( {x + 1} \right)^2} + {y^2} \ge 0,\forall x,y\\ g)4\left( {x - 2} \right)\left( {x - 1} \right)\left( {x + 4} \right)\left( {x + 8} \right) + 25{x^2} \end{array}$