1, x²-6x+10=(x²-6x+9)+1=(x-3)²+1 >0 ∀x∈R
2, x²-8x+19=(x²-8x+16)+3=(x-4)²+3 >0 ∀x∈R
3, 3x²-6x+5=3(x²-2x+1-1+$\frac{5}{3}$ )=3[(x-1)²+$\frac{2}{3}$ ]
= 3(x-1)² +2 >0 ∀x∈R
4, x²+x+1=x²+2.x.1/2+(1/2)²-(1/2)²+1=(x+1/2)²+3/4 >0 ∀x∈R
5. (x-3)(x-5)+44= x²-8x+15+44=(x²-8x+16)+43
= (x-4)²+43 >0 ∀x∈R
6, x²+10x+27=(x²+10x+25)+2=(x+5)²+2 >0 ∀x∈R