Question 700998
You probably mean f(x)=(x^3+x^2-4x-4)/(x^2+2x-3) ,
which I can write as
{{{f(x)=(x^3+x^2-4x-4)/(x^2+2x-3)}}}
 
Factoring polynomials never goes away. You have to factor numerator and denominator.
 
{{{x^3+x^2-4x-4}}} can be factored "by grouping" as
{{{x^3+x^2-4x-4=(x^3-4x)+(x^2-4)=x(x^2-4)+(x^2-4)=(x+1)(x^2-4)}}}
It can be factored further because the difference of squares {{{x^2-4=x^2-2^2}}}
factors as {{{(x-2)(x+2)}}}
 
{{{x^2+2x-3=(x+3)(x-1)}}}
 
Putting it all together:
{{{f(x)=(x^3+x^2-4x-4)/(x^2+2x-3)=(x+1)(x+2)(x-2)/((x+3)(x-1))}}}
The function is not defined (it does not exist) for {{{x=1}}} and for {{{x=-3}}} because the denominator is zero for those values of {{{x}}}.
To be continuous, the function has to be defined. So at those points the function is not continuous.
At {{{x=1}}} and at {{{x=-3}}} , the function has a vertical asymptote.
As {{{x}}} approaches {{{x=1}}} from either side, the denominator approaches zero.
At the same time, the numerator approaches {{{(1+1)(1+2)(1-2)=-6}}}.
As a consequence the absolute value of the function grows without limits,
and the graph hugs the vertical line {{{x=1}}}.
Something similar happens at {{{x=-3}}}.
{{{graph(300,300,-7,5,-20,20,(x^3+x^2-4x-4)/(x^2+2x-3))}}}
 
NOTE: I would call those values of {{{x}}} , {{{x=1}}} and {{{x=-3}}} , points of discontinuity.
However, names of different kinds of discontinuity are not universally agreed upon,
and some may not like to call {{{x=1}}} and {{{x=-3}}} points of discontinuity.
They would say that {{{x=1}}} and {{{x=-3}}} are not really points, with an x value and a y value, and the name "points of discontinuity" could be confused with "point discontinuity," which is a different kind of discontinuity.
A function may not be continuous at one point that is just a hole in the graph,
as in {{{g(x)=(x+1)/(x+1)}}}.
That function has {{{g(x)=1}}} for all values of {{{x}}} except {{{x=-1}}}, where g(x) is not defined.
The function {{{g(x)=(x+1)/(x+1)}}} graphs as a horizontal line minus the point (-1,1) that is a hole in the graph.
Some call that a "point discontinuity", and others call it a "removable discontinuity".