\[\frac{b}{b + 1} - \frac{b^{2}}{b^{2} - 1} = \frac{b(b - 1) - b^{2}}{b^{2} - 1} =\]
\[= \frac{b^{2} - 1 - b^{2}}{b^{2} - 1} = - \frac{1}{b^{2} - 1}.\]