Let's determine [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] where [tex]\(f(x) = x^4 - x^3 + x^2\)[/tex] and [tex]\(g(x) = -x^2\)[/tex].
Given [tex]\(f(x) = x^4 - x^3 + x^2\)[/tex] and [tex]\(g(x) = -x^2\)[/tex]:
1. We need to find the quotient [tex]\(\frac{f(x)}{g(x)}\)[/tex]: [tex]\[
\frac{f(x)}{g(x)} = \frac{x^4 - x^3 + x^2}{-x^2}
\][/tex]
2. Split the numerator and divide each term by the denominator: [tex]\[
\frac{x^4}{-x^2} - \frac{x^3}{-x^2} + \frac{x^2}{-x^2}
\][/tex]
3. Simplify each term individually: [tex]\[
\frac{x^4}{-x^2} = -x^2
\][/tex] [tex]\[
\frac{x^3}{-x^2} = -x
\][/tex] [tex]\[
\frac{x^2}{-x^2} = -1
\][/tex]
4. Combine the simplified terms: [tex]\[
-x^2 - x - 1
\][/tex]
Therefore, the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] simplifies to: [tex]\[
-x^2 + x - 1
\][/tex]
Comparing with the given options: 1. [tex]\(x^2 - x + 1\)[/tex] 2. [tex]\(x^2 + x + 1\)[/tex] 3. [tex]\(-x^2 + x - 1\)[/tex] 4. [tex]\(-x^2 - x - 1\)[/tex]
The correct answer is: [tex]\[
\boxed{-x^2 + x - 1}
\][/tex]
