Sure, let's solve this step by step.
Given the functions:
[tex]\[ u(x) = x^5 - x^4 + x^2 \][/tex]
[tex]\[ v(x) = -x^2 \][/tex]
We need to find the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex].
First, we write the fraction [tex]\(\frac{u(x)}{v(x)}\)[/tex]:
[tex]\[ \frac{u(x)}{v(x)} = \frac{x^5 - x^4 + x^2}{-x^2} \][/tex]
Next, we simplify this expression by dividing each term in the numerator by the denominator [tex]\(-x^2\)[/tex]:
1. Divide [tex]\(x^5\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[ \frac{x^5}{-x^2} = -x^{5-2} = -x^3 \][/tex]
2. Divide [tex]\(x^4\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[ \frac{x^4}{-x^2} = -x^{4-2} = -x^2 \][/tex]
3. Divide [tex]\(x^2\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[ \frac{x^2}{-x^2} = -1 \][/tex]
Combining these results, we get:
[tex]\[ \frac{x^5 - x^4 + x^2}{-x^2} = -x^3 + x^2 - 1 \][/tex]
Therefore, the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] is:
[tex]\[ -x^3 + x^2 - 1 \][/tex]
Among the given options, the correct one is:
[tex]\[ \boxed{-x^3 + x^2 - 1} \][/tex]
