a) \(2x^2+8x+15\) \(=2\left(x^2+4x+\frac{15}{2}\right)\) \(=2\left(x^2+4x+4+\frac{7}{2}\right)=2\left(x+2\right)^2+7\ge7>0\)
b) \(-x^2+x-3=-\left(x^2-x+3\right)\) \(=-\left(x^2-2\cdot x\cdot\frac{1}{2}+\frac{1}{4}+\frac{11}{4}\right)=-\frac{11}{4}-\left(x-\frac{1}{2}\right)^2< 0\)
c) \(-4x^2+8x-11=-\left(4x^2-8x+11\right)\) \(=-\left(4x^2-2\cdot2\cdot2x+4+7\right)=-7-\left(2x-2\right)^2< 0\)
d) \(-9x^2+12x-15=-\left(9x^2-12x+15\right)\) \(=-\left(9x^2-2\cdot3x\cdot2+4+11\right)=-11-\left(3x-2\right)^2< 0\)
