To simplify the expression [tex]\(\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}\)[/tex], let's go through a systematic step-by-step process:
1. Factoring the numerator:
The numerator is [tex]\(x^2 + x - 2\)[/tex]. We need to factor this quadratic expression.
The factors of [tex]\(-2\)[/tex] (constant term) that add up to [tex]\(1\)[/tex] (coefficient of [tex]\(x\)[/tex]) are [tex]\(2\)[/tex] and [tex]\(-1\)[/tex]. Therefore, the numerator can be factored as:
[tex]\[
x^2 + x - 2 = (x + 2)(x - 1)
\][/tex]
2. Factoring the denominator:
The denominator is a cubic polynomial [tex]\(x^3 - x^2 + 2x - 2\)[/tex]. We can try to factor this by grouping.
By grouping terms:
[tex]\[
x^3 - x^2 + 2x - 2 = x^2(x - 1) + 2(x - 1)
\][/tex]
Notice that [tex]\((x - 1)\)[/tex] is a common factor:
[tex]\[
x^3 - x^2 + 2x - 2 = (x - 1)(x^2 + 2)
\][/tex]
3. Simplifying the fraction:
Now, substitute the factored forms into the original fraction:
[tex]\[
\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2} = \frac{(x + 2)(x - 1)}{(x - 1)(x^2 + 2)}
\][/tex]
We can cancel the common factor [tex]\((x - 1)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{(x + 2)(x - 1)}{(x - 1)(x^2 + 2)} = \frac{x + 2}{x^2 + 2}
\][/tex]
So, the simplest form of the expression is:
[tex]\[
\frac{x + 2}{x^2 + 2}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{x+2}{x^2+2}}
\][/tex]
This corresponds to option [tex]\(C\)[/tex].
