Answer
$$\eqalign{
& \left( {\text{a}} \right) - \infty ,\infty \cr
& \left( {\text{b}} \right) - x - 2{\text{ is the slant asymptote}} \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{{1 + x - 2{x^2} - {x^3}}}{{{x^2} + 1}} \cr
& \left( {\text{a}} \right){\text{ Calculate }}\mathop {\lim }\limits_{x \to \infty } f\left( x \right){\text{ and }}\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) \cr
& \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \frac{{1 + x - 2{x^2} - {x^3}}}{{{x^2} + 1}} \cr
& = \frac{{\overbrace {\mathop {\lim }\limits_{x \to \infty } \left( {1 + x - 2{x^2} - {x^3}} \right)}^{{\text{arbitrary large to negative infinite }}}}}{{\underbrace {\mathop {\lim }\limits_{x \to \infty } \left( {{x^2} + 1} \right)}_{{\text{arbitrary large to positive infinite }}}}} = - \infty \cr
& and \cr
& \mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to - \infty } \frac{{1 + x - 2{x^2} - {x^3}}}{{{x^2} + 1}} \cr
& = \frac{{\overbrace {\mathop {\lim }\limits_{x \to - \infty } \left( {1 + x - 2{x^2} - {x^3}} \right)}^{{\text{arbitrare large to positive infinite }}}}}{{\underbrace {\mathop {\lim }\limits_{x \to - \infty } \left( {{x^2} + 1} \right)}_{{\text{arbitrare large to positive infinite }}}}} = \infty \cr
& \cr
& \left( {\text{b}} \right){\text{Using the long division}}{\text{, the function }}f\left( x \right){\text{ can be written as}} \cr
& f\left( x \right) = \frac{{1 + x - 2{x^2} - {x^3}}}{{{x^2} + 1}} = \overbrace { - x - 2}^{\ell \left( x \right)} + \frac{{2x + 3}}{{{x^2} + 1}} \cr
& {\text{As }}x{\text{ tends to positive infinite or negative infinite}}{\text{, }} \cr
& {\text{the term }}\frac{{2x + 3}}{{{x^2} + 1}}{\text{ approaches to 0}} \cr
& {\text{and }}\mathop {\lim }\limits_{x \to \infty } \ell \left( x \right) = - \infty ,{\text{ }}\mathop {\lim }\limits_{x \to - \infty } \ell \left( x \right) = + \infty ,{\text{ then}} \cr
& {\text{The line }}\ell \left( x \right) = - x - 2{\text{ is a slant asymptote}}{\text{.}} \cr} $$
