To find the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex], we need to divide the function [tex]\(u(x)\)[/tex] by the function [tex]\(v(x)\)[/tex].
Given [tex]\(u(x) = x^5 - x^4 + x^2\)[/tex] and [tex]\(v(x) = -x^2\)[/tex], we will perform the division step-by-step.
1. Write the division in fraction form:
[tex]\[
\left(\frac{u}{v}\right)(x) = \frac{x^5 - x^4 + x^2}{-x^2}
\][/tex]
2. Divide each term in the numerator by the denominator [tex]\( -x^2 \)[/tex]:
[tex]\[
\frac{x^5}{-x^2} - \frac{x^4}{-x^2} + \frac{x^2}{-x^2}
\][/tex]
3. Simplify each term:
[tex]\[
\frac{x^5}{-x^2} = -x^{5-2} = -x^3
\][/tex]
[tex]\[
\frac{x^4}{-x^2} = -x^{4-2} = -x^2
\][/tex]
[tex]\[
\frac{x^2}{-x^2} = -1
\][/tex]
4. Combine the simplified terms:
[tex]\[
\left(\frac{u}{v}\right)(x) = -x^3 + x^2 - 1
\][/tex]
Thus, the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] is:
[tex]\[
-x^3 + x^2 - 1
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-x^3 + x^2 - 1}
\][/tex]
