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=1/x(x+1)-1/(x+1)(x+2)+1/(x+1)^2-1/(x+1)(x+3)
=(x+2)/x(x+1)(x+2)-x/x(x+1)(x+2)+(x+3)/(x+1)^2(x+3)-(x+1)/(x+1)^2(x+3)
=2/ x(x+1)(x+2)+2/(x+1)^2(x+3)
=2(x+1)(x+3) /x(x+1)^2 (x+2) (x+3)+2x(x+2)/ x(x+1)^2 (x+2) (x+3)
=[2(x+1)(x+3)-2x(x+2)]/ x(x+1)^2 (x+2) (x+3)
=(4x+6)/ x(x+1)^2 (x+2) (x+3)
=(x+2)/x(x+1)(x+2)-x/x(x+1)(x+2)+(x+3)/(x+1)^2(x+3)-(x+1)/(x+1)^2(x+3)
=2/ x(x+1)(x+2)+2/(x+1)^2(x+3)
=2(x+1)(x+3) /x(x+1)^2 (x+2) (x+3)+2x(x+2)/ x(x+1)^2 (x+2) (x+3)
=[2(x+1)(x+3)-2x(x+2)]/ x(x+1)^2 (x+2) (x+3)
=(4x+6)/ x(x+1)^2 (x+2) (x+3)
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