1/(a+1)-1/(a+2)+1/(a+3)-1/(a+4)
=1/(a+1)-1/(a+4)-[1/(a+2)-1/(a+3)]
=(a+4-a-1)/[(a+1)(a+4)]-(a+3-a-2)/[(a+2)(a+3)]
=3/[(a+1)(a+4)]-1/[(a+2)(a+3)]
=[3(a²+5a+6)-(a²+5a+4)]/[(a+1)(a+4)(a+2)(a+3)]
=(2a²+10a+14)/[(a+1)(a+4)(a+2)(a+3)]
=2(a²+5a+7)/[(a+1)(a+4)(a+2)(a+3)]