To determine the simplest form of the given expression
[tex]\[
\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2},
\][/tex]
we need to simplify the fraction and check which of the provided options it matches. Here is the step-by-step procedure:
1. Factor the Numerator and Denominator:
We start by factoring both the numerator and the denominator if possible.
The numerator [tex]\(x^2 + x - 2\)[/tex] can be factored as:
[tex]\[
x^2 + x - 2 = (x + 2)(x - 1)
\][/tex]
The denominator [tex]\(x^3 - x^2 + 2x - 2\)[/tex] can be factored by grouping:
[tex]\[
x^3 - x^2 + 2x - 2 = x^2(x - 1) + 2(x - 1) = (x^2 + 2)(x - 1)
\][/tex]
2. Simplify the Expression:
Now we have:
[tex]\[
\frac{(x + 2)(x - 1)}{(x^2 + 2)(x - 1)}
\][/tex]
We can cancel the common term [tex]\((x - 1)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{x + 2}{x^2 + 2}
\][/tex]
3. Compare with the Provided Options:
Let's see which option matches [tex]\(\frac{x+2}{x^2+2}\)[/tex]:
- [tex]\( \text{Option A:} \frac{1}{x-2} \)[/tex]
This is not equivalent to [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].
- [tex]\( \text{Option B:} \frac{1}{x+2} \)[/tex]
This is not equivalent to [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].
- [tex]\( \text{Option C:} \frac{x+2}{x^2+2} \)[/tex]
This is exactly equivalent to our simplified form [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].
- [tex]\( \text{Option D:} \frac{x-1}{x^2+2} \)[/tex]
This is not equivalent to [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex].
Thus, the simplest form of the given expression is
[tex]\[
\boxed{ \frac{x+2}{x^2+2} },
\][/tex]
which corresponds to option C.
