find straight lines

observation x=1 is a root

so there will be roots (x-1)(x-2)(x-3)=0

so 3 lines parallel to the y axis.

There is a method for solving cubics.. infact a formula.. but i guess i will die before i am able to learn it..

As for the observation, iwould try to look at the above one as a perfect square.. :

so i would have seen 6x²-12x+6 + x in the given expression

x³-6x²+11x-6=x³- (6x²-12x+6 + x) = x³-x - 6(x-1)²

The rest you can solve..

Another thing is that if you have to guess a root, and if that root happens to be an integer, then there is a rule (which is easily proved) that the roots of a polynomial (if integer) are factors of the constant term..

so the root will be a factor of 6 which means 1, 2, 3, -1, -2, -3

so the roots have to be one of these {1, 2, 3, -1, -2, -3 }