因为[(x-2)^3-(x-1)^2+1]/(x-2)
=[(x-2)^3/(x-2)]-[(x-1)^2-1]/(x-2)
=(x-2)^2-[(x-1+1)(x-1-1)/(x-2)]
=x^2-4x+4-x
=x^2-5x+4,
又因为x^2-5x-1997=0,
所以x^2-5x=1997,
所以原式=[(x-2)^3-(x-1)^2+1]/(x-2)
=x^2-5x+4
=1997+4=2001.