Answer
$x^{2}+1$
Work Step by Step
Simplifying Rational Expressions
1. Factor the numerator and the denominator completely.
2. Divide both the numerator and the denominator by any common factors.
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Numerator: factor in pairs
$x^{3}-2x^{2}+x-2=x^{2}(x-2)+(x-2)$
$=(x-2)(x^{2}+1)$
Denominator: fully factored.
Expression = $\displaystyle \frac{(x-2)(x^{2}+1)}{(x-2)}$
... divide both with the common factor: $(x-2)$
Expression = $\displaystyle \frac{x^{2}+1}{1} = x^{2}+1$
