Answer
$$ - \frac{1}{{\left( {{x^2} + 1} \right)\sqrt {{x^2} + 1} }}$$
Work Step by Step
$$\eqalign{
& \frac{{\left( {{x^2} - 1} \right)\sqrt {{x^2} + 1} - \frac{{{x^4}}}{{\sqrt {{x^2} + 1} }}}}{{{x^2} + 1}} \cr
& {\text{Multiply the numerator and denominator by }}\sqrt {{x^2} + 1} \cr
& = \frac{{\left( {{x^2} - 1} \right){{\left( {\sqrt {{x^2} + 1} } \right)}^2} - \frac{{{x^4}\sqrt {{x^2} + 1} }}{{\sqrt {{x^2} + 1} }}}}{{\left( {{x^2} + 1} \right)\sqrt {{x^2} + 1} }} \cr
& = \frac{{\left( {{x^2} - 1} \right)\left( {{x^2} + 1} \right) - {x^4}}}{{\left( {{x^2} + 1} \right)\sqrt {{x^2} + 1} }} \cr
& {\text{Simplify}} \cr
& = \frac{{{x^4} - 1 - {x^4}}}{{\left( {{x^2} + 1} \right)\sqrt {{x^2} + 1} }} \cr
& = - \frac{1}{{\left( {{x^2} + 1} \right)\sqrt {{x^2} + 1} }} \cr} $$
