To simplify the expression [tex]\(\frac{x^2+x-2}{x^3-x^2+2x-2}\)[/tex], let's go through the steps.
### Step 1: Factor the numerator
The numerator is [tex]\(x^2 + x - 2\)[/tex]. To factor this quadratic expression, we need to find two numbers that multiply to [tex]\(-2\)[/tex] (the constant term) and add up to [tex]\(1\)[/tex] (the coefficient of the middle term [tex]\(x\)[/tex]).
The two numbers that satisfy this condition are [tex]\(2\)[/tex] and [tex]\(-1\)[/tex]:
[tex]\[
x^2 + x - 2 = (x + 2)(x - 1)
\][/tex]
### Step 2: Factor the denominator
The denominator is [tex]\(x^3 - x^2 + 2x - 2\)[/tex]. We can try to factor by grouping:
[tex]\[
x^3 - x^2 + 2x - 2 = (x^3 - x^2) + (2x - 2)
\][/tex]
Notice that we can factor [tex]\(x^2\)[/tex] out of the first group and [tex]\(2\)[/tex] out of the second group:
[tex]\[
x^3 - x^2 + 2x - 2 = x^2(x - 1) + 2(x - 1)
\][/tex]
Now, we can factor [tex]\((x - 1)\)[/tex] out of the entire expression:
[tex]\[
x^3 - x^2 + 2x - 2 = (x - 1)(x^2 + 2)
\][/tex]
### Step 3: Simplify the expression
We have the factored form of both the numerator and the denominator:
[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 out the common factor [tex]\((x - 1)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{(x + 2)(x - 1)}{(x - 1)(x^2 + 2)} = \frac{x + 2}{x^2 + 2}
\][/tex]
Thus, the expression in its simplest form is:
[tex]\[
\frac{x + 2}{x^2 + 2}
\][/tex]
The correct answer is:
D. [tex]\(\frac{x + 2}{x^2 + 2}\)[/tex]
