[(x+2)/(x+3) - (x+1)/(x+2) + (x+4)/(x+5) - (x+3)/(x+4)] / (x^2 + 7x + 13)/(x^2 + 8x + 15)
=[-1/(x+3)+1/(x+2)-1/(x+5)+1/(x+4)] / (x^2 + 7x + 13)/(x^2 + 8x + 15)
=[(1/(x+4)-1/(x+3))+(1/(x+2)-1/(x+5))]/ (x^2 + 7x + 13)/(x^2 + 8x + 15)
=(-1/(x^2+7x+12)+3/(x^2+7x+10))/ (x^2 + 7x + 13)/(x^2 + 8x + 15)
=[(2x^2+14x+26)/(x+2)(x+3)(x+4)(x+5)]/ [(x^2 + 7x + 13)/(x+3)(x+5)]
=2/(x+2)(x+4)