Factorise: $x^{3}-2x^{2}-x+2$

(A) Given expression: $x^{3}-2x^{2}-x+2$
Step $1$: Group the terms to facilitate factoring by grouping.
$x^{3}-2x^{2}-x+2 = (x^{3}-2x^{2}) - (x-2)$
Step $2$: Factor out the common terms from each group.
$= x^{2}(x-2) - 1(x-2)$
Step $3$: Factor out the common binomial $(x-2)$.
$= (x^{2}-1)(x-2)$
Step $4$: Use the identity $a^{2}-b^{2} = (a-b)(a+b)$ to factor $(x^{2}-1)$.
$= (x-1)(x+1)(x-2)$
Thus,the factors are $(x-1)(x+1)(x-2)$.