Note that a^2+5a+6 factorises: (a+3)(a+2). 4-a^2 also factorises: (2-a)(2+a). a^2+a-6 factorises, too: (a+3)(a-2).
Now we can simplify things a bit. (a+2)/((a+3)(a+2))=1/(a+3). (2+a)/((2-a)(2+a))=1/(2-a). (2-a)/((a+3)(a-2)=-(a-2)/((a+3)(a-2))=-1/(a+3), which cancels out 1/(a+3) calculated earlier: 1/(a+3)-1/(2-a)-1/(a+3)=-1/(2-a)=1/(a-2).