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

Slant asymptote: ##y = x-1##. See the -inclusive graph.

By division,

##y = f(x)= x-1+1/(x+4)##

Rearranged,

##(y-x+1)(x+4)=1##, revealing that the graph is a hyperbola with

asymptotes

##(y-x+1)(x+4)=0##.

Separately,

y-x+1=0 gives a slant asymptote and

x+4=0 gives a vertical asymptote.

graph{((y-x+1)(x+4)-1)(y-x+1)(x+0.000001y+4)=0 [-23, 23.02, -12.7, 12.6]}