Let [tex]\( f(x) \)[/tex] be defined as:
[tex]\[ f(x) = x^4 - x^3 + x^2 \][/tex]
Let [tex]\( g(x) \)[/tex] be defined as:
[tex]\[ g(x) = -x^2 \][/tex]
We are asked to find the expression for [tex]\(\left( \frac{f}{g} \right)(x)\)[/tex].
First, we will divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \][/tex]
[tex]\[ = \frac{x^4 - x^3 + x^2}{-x^2} \][/tex]
We can separate the terms in the numerator and divide each by [tex]\( -x^2 \)[/tex]:
[tex]\[ = \frac{x^4}{-x^2} - \frac{x^3}{-x^2} + \frac{x^2}{-x^2} \][/tex]
Simplify each term:
[tex]\[ = -x^2 + x - 1 \][/tex]
So, we find:
[tex]\[ \left( \frac{f}{g} \right)(x) = -x^2 + x - 1 \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{-x^2 + x - 1} \][/tex]
