\(A=x^2-x+1\)
\(=x^2-2\cdot x\cdot\frac12+\frac14+\frac34\)
\(=\left(x-\frac12\right)^2+\frac34\ge\frac34>0\forall x\)
\(B=x^2+x+1\)
\(=x^2+x+\frac14+\frac34\)
\(=\left(x+\frac12\right)^2+\frac34\ge\frac34>0\forall x\)
\(C=x^2+2x+2\)
\(=x^2+2x+1+1\)
\(=\left(x+1\right)^2+1\ge1>0\forall x\)