Question 1115363
Some factorization will help to identify some of those details.


{{{(x*x^2)/(x*(2x^2-x-3))}}}


{{{highlight_green((x/x)((x^2)/(2x-3)(x+1)))}}}


Undefined for x at 0, but since there is {{{x/x}}}, a HOLE at x = 0.


Notice degree of numerator and denominator of g(x) are the same, so horizontal asymptote is {{{y=1/2}}}.


Vertical asymptotes:  Look at the denominator of the factored form.
Asymptotes for {{{x=-1}}} and for {{{x=3/2}}}.