Question 855986
This will help most of the way, useful in sketching a graph.


Polynomial Division will show quotient of {{{x+1+(2x+4)/(x^2-3x+2)}}}, so a slant asymptote will be {{{highlight(y=x+1)}}}, since the remainder, {{{(2x+4)/(x^2-3x+2)}}} will become increasingly smaller as x becomes unbounded.


f(x) is factorable.
{{{highlight_green(f(x)=((x+1)(x-2)(x-3))/((x-1)(x-2)))}}}


Numerator factored indicates zeros at x=-1 and x=3.
Hole at x=2.  (see common factor in numerator and denominator).
Vertical Asymptote at x=1 (see what this does to denominator).
Slant asymptote, y=x+1.
You can test for signs around critical points -1,1,2,3.


{{{graph(300,300,-3,6,-10,15,(x^3-4x^2+x+6)/(x^2-3x+2))}}}

Not sure why the hole at x=2 does not appear as a hole, but that IS a hole, undefined function at x=2.