Question

select the correct answer.
what is this expression in simplest form?
\\(\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}\\)
a. \\(\frac{1}{x - 2}\\)
b. \\(\frac{1}{x + 2}\\)
c. \\(\frac{x - 1}{x^2 + 2}\\)
d. \\(\frac{x + 2}{x^2 + 2}\\)

Explanation:

Step1: Factor the denominator

First, factor the denominator \(x^2 - x + 2x - 2\). Combine like terms: \(x^2 + x - 2\). Then factor the quadratic: \(x^2 + x - 2=(x + 2)(x - 1)\)? Wait, no, wait. Wait, the original denominator is \(x^2 - x + 2x - 2\)? Wait, maybe a typo? Wait, the numerator is \(x^2 + x - 2\). Let's factor numerator and denominator.

Numerator: \(x^2 + x - 2\). Factor it: find two numbers that multiply to -2 and add to 1. Those numbers are 2 and -1. So \(x^2 + x - 2=(x + 2)(x - 1)\).

Denominator: Wait, the original expression is \(\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}\)? Wait, maybe the denominator is \(x^3 - x^2 + 2x - 2\). Let's factor the denominator by grouping. Group terms: \((x^3 - x^2)+(2x - 2)=x^2(x - 1)+2(x - 1)=(x - 1)(x^2 + 2)\).

Wait, no, maybe the denominator is \(x^2 - x + 2x - 2\) which is \(x^2 + x - 2\)? No, the user's image shows the denominator as \(x^3 - x^2 + 2x - 2\)? Wait, maybe I misread. Let's check the options. The options have denominators like \(x - 2\), \(x + 2\), \(x^2 + 2\), etc.

Wait, let's re-express the original expression: \(\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}\). Let's factor numerator and denominator.

Numerator: \(x^2 + x - 2=(x + 2)(x - 1)\) (as before).

Denominator: Factor by grouping: \(x^3 - x^2 + 2x - 2 = x^2(x - 1) + 2(x - 1) = (x - 1)(x^2 + 2)\).

So the expression becomes \(\frac{(x + 2)(x - 1)}{(x - 1)(x^2 + 2)}\). Cancel out the common factor \((x - 1)\) (assuming \(x
eq 1\)): \(\frac{x + 2}{x^2 + 2}\). Wait, but option D is \(\frac{x + 2}{x^2 + 2}\)? Wait, the options:

A. \(\frac{1}{x - 2}\)

B. \(\frac{1}{x + 2}\)

C. \(\frac{x - 1}{x^2 + 2}\)

D. \(\frac{x + 2}{x^2 + 2}\)

Wait, maybe I made a mistake in factoring the numerator. Wait, numerator is \(x^2 + x - 2\), which factors to \((x + 2)(x - 1)\). Denominator: let's check again. If the denominator is \(x^3 - x^2 + 2x - 2\), factoring by grouping: \(x^2(x - 1) + 2(x - 1) = (x - 1)(x^2 + 2)\). So cancel \((x - 1)\), we get \(\frac{x + 2}{x^2 + 2}\), which is option D. But the user's selected option is D, which is correct? Wait, maybe the original expression's numerator is \(x^2 + x - 2\) and denominator is \(x^3 - x^2 + 2x - 2\). So after factoring, we get \(\frac{x + 2}{x^2 + 2}\), which is option D.

Wait, maybe I misread the numerator. If the numerator is \(x^2 + x - 2\) and denominator is \(x^3 - x^2 + 2x - 2\), then factoring gives \(\frac{(x + 2)(x - 1)}{(x - 1)(x^2 + 2)} = \frac{x + 2}{x^2 + 2}\), which is option D. So the correct answer is D.

Step2: Simplify the expression

After factoring numerator and denominator and canceling common factors, we get \(\frac{x + 2}{x^2 + 2}\), which matches option D.

Answer:

D. \(\frac{x + 2}{x^2 + 2}\)