\(B=\frac{4x+2}{2\left(x^2+2\right)}=\frac{-\left(x^2+2\right)}{2\left(x^2+2\right)}+\frac{x^2+4x+4}{x^2+2}=-\frac{1}{2}+\frac{\left(x+2\right)^2}{x^2+2}\ge-\frac{1}{2}\)
\(B=\frac{x^2+2}{x^2+2}-\frac{x^2-2x+1}{x^2+2}=1-\frac{\left(x-1\right)^2}{x^2+2}\le1\)
\(C=\frac{-\left(x^2+1\right)}{x^2+1}+\frac{x^2+4x+4}{x^2+1}=-1+\frac{\left(x+2\right)^2}{x^2+1}\ge-1\)
\(C=\frac{4x^2+4}{x^2+1}-\frac{4x^2-4x+1}{x^2+1}=4-\frac{\left(2x-1\right)^2}{x^2+1}\le4\)
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