a) $A=x^2-10x+27 $
$=(x-5)^2 + 2 > 0 $
b) $B=x^2+9x+24 $
$=\bigg(x+\dfrac{9}{2}\bigg)^2 + \dfrac{15}{4} > 0 $
c) $C =4x-x^2-8 $
$=-(x^2-4x+4) - 12 $
$=-(x+2)^2 - 12 < 0 $
d) $D = 6x-2x^2-11 $
$=-2.(x^2-3x + \dfrac{9}{4} )^2 -\dfrac{31}{2} < 0 $
e) $E =2x^2-4xy-6x+5y^2+2y+13$
$=(x-2y)^2 + (x-3)^2 + (y+1)^2+3> 0 $
g) $G = (x-1)(x+2)(x-5)(x+4) +15$
$=(x-1)(x-5)(x+2)(x+4) + 15$
$=(x^2-6x+5).(x^2-6x+8) + 15 $
$=(x^2-6x+5)^2 + 3.(x^2+6x+5) + \dfrac{9}{4} +\dfrac{51}{4} $
$=(x^2-6x+\dfrac{17}{3})^2 + \dfrac{51}{4} > 0 $
