Answer
$\frac{-2}{(x+2)(x+3)}$,
See graph.
Work Step by Step
Step 1. Factor the denominators and perform the subtraction to get:
$f(x)=\frac{2}{(x+1)(x+2)}-\frac{4}{(x+1)(x+3)}=\frac{2(x+3)-4(x+2)}{(x+1)(x+2)(x+3)}=\frac{-2x-2}{(x+1)(x+2)(x+3)}=\frac{-2}{(x+2)(x+3)}$
with a hole at $(-1,-1)$ and $x\ne-3,-2$
Step 2. We can identify two vertical asymptotes as $x=-3$ and $x=-2$
Step 3. We can also identify a horizontal asymptote as $y=0$ and a y-intercept as $y=-\frac{1}{3}$
Step 4. Testing signs across the vertical asymptotes, we have
$...(-)...(-3)...(+)...(-2)...(-)...$
Step 5. Based on the above results, we can graph the function as shown in the figure.
