How do you find the slant asymptote of y=(x^3 - 4x^2 + 2x -5)/ (x^2 + 2)?

$\textcolor{b l u e}{\text{Slant Asymptote}}$
$\textcolor{b l u e}{y = x - 4}$

Explanation:

To find the slant asymptote, we divide ${x}^{3} - 4 {x}^{2} + 2 x - 5$ by ${x}^{2} + 2$

The resulting quotient not including the remainder part represents the slant asymptote

Let us divide

" " " " " " " " " " " "underline(x-4" " " " " " " " " " " " " ")
${x}^{2} + 0 \cdot x + 2 \lceiling {x}^{3} - 4 {x}^{2} + 2 x - 5$
" " " " " " " " " " " "underline(x^3+0x^2+2x" " " " " " " ")
$\text{ " " " " " " " " " " " " } - 4 {x}^{2} + 0 - 5$
" " " " " " " " " " " " " "underline(-4x^2+0x-8" " " " " ")
$\text{ " " " " " " " " " " " " " " " " " " " "" " " } + 3$

Observe the quotient $x - 4$ so that our slant asymptote is

$y = x - 4$

Kindly see the graph of $y = \frac{{x}^{3} - 4 {x}^{2} + 2 x - 5}{{x}^{2} + 2} \text{ }$(colored red) and the slant asymptote $y = x - 4 \text{ }$(colored blue). God bless....I hope the explanation is useful.