Question 951126
"where will the/a graph come from?"  The question does not mean what you discussed. 


Look at factorization.


{{{4(2-x^2)/(x+2)^2}}}


{{{-4(x^2-2)/(x+2)^2}}}


Two irrational zeros, meaning crosses the x axis in two places.
{{{x^2=2}}}
{{{x=-sqrt(2)}}} and at {{{x=sqrt(2)}}}.


Vertical asymptote at {{{x=-2}}}.  The function is undefined at x=-2.


For unbounded x left or right, the function will approach {{{-4x^2/x^2}}} or -4, which means the line {{{y=-4}}} is the horizontal asymptote.



?
Something is not right about this analysis or the graphs.....
{{{graph(300,300,-12,12,-12,12,(8x-4x^2)/(x+2)^2)}}}


{{{graph(300,300,-12,12,-22,2,(8x-4x^2)/(x+2)^2)}}}


{{{graph(300,300,-32,32,-22,22,(8x-4x^2)/(x+2)^2)}}}


{{{graph(300,300,-4,4,-4,4,(8x-4x^2)/(x+2)^2)}}}