To determine the function [tex]\(\frac{f(x)}{g(x)}\)[/tex] where
[tex]\[ f(x) = x^4 - x^3 + x^2 \][/tex]
and
[tex]\[ g(x) = -x^2 \][/tex]
we need to perform the division of [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex].
First, write the expression for [tex]\(\frac{f(x)}{g(x)}\)[/tex]:
[tex]\[
r(x) = \frac{f(x)}{g(x)} = \frac{x^4 - x^3 + x^2}{-x^2}
\][/tex]
Now, simplify this expression by dividing each term in the numerator by the denominator:
[tex]\[
r(x) = \frac{x^4}{-x^2} - \frac{x^3}{-x^2} + \frac{x^2}{-x^2}
\][/tex]
Perform the division for each term:
[tex]\[
r(x) = -x^{4-2} + x^{3-2} - 1
\][/tex]
[tex]\[
r(x) = -x^2 + x - 1
\][/tex]
Thus, the simplified form of the function [tex]\(\frac{f(x)}{g(x)}\)[/tex] is:
[tex]\[
r(x) = -x^2 + x - 1
\][/tex]
So the correct answer is:
[tex]\[
- x^2 + x - 1
\][/tex]
