Answer
$\frac{x-2}{x+2}$; see graph
Work Step by Step
Step 1. Multiply the numerator and the denominator with $(x+2)(x-2)$. We have
$f(x)=\frac{x^2-4-3(x-2)}{x^2-4+x+2}=\frac{x^2-3x+2}{x^2+x-2}=\frac{(x-1)(x-2)}{(x+2)(x-1)}=\frac{x-2}{x+2}$
where $x\ne 1, \pm2,$ with holes at $(1, -\frac{1}{3})$ and $(2,0)$
Step 2. We can identify a vertical asymptote as $x=-2$
Step 3. We can also identify a horizontal asymptote as $y=1$
Step 4. We can find the y-intercept at $y=-1$
Step 5. Testing signs across the vertical asymptotes, we have
$...(+)...(-2)...(-)...$
Step 6. Based on the above results, we can graph the function as shown in the figure.
