If \( u(x)=x^{5}-x^{4}+x^{2} \) and \( v(x)=-x^{2} \), which expression is equivalent to \( \left(\frac{u}{v}\right)(x) \) ? \( x^{3}-x^{2} \) \( -x^{3}+x^{2} \) \( -x^{3}+x^{2}-1 \) \( x^{3}-x^{2}+1 \)

Simplify the expression by following steps: - step0: Divide the terms: \(\frac{x^{5}-x^{4}+x^{2}}{-x^{2}}\) - step1: Rewrite the fraction: \(-\frac{x^{5}-x^{4}+x^{2}}{x^{2}}\) - step2: Factor: \(-\frac{x^{2}\left(x^{3}-x^{2}+1\right)}{x^{2}}\) - step3: Reduce the fraction: \(-\left(x^{3}-x^{2}+1\right)\) - step4: Calculate: \(-x^{3}+x^{2}-1\) To find the expression equivalent to \( \left(\frac{u}{v}\right)(x) \), we start with the given functions: 1. \( u(x) = x^{5} - x^{4} + x^{2} \) 2. \( v(x) = -x^{2} \) Now, we compute \( \frac{u(x)}{v(x)} \): \[ \frac{u(x)}{v(x)} = \frac{x^{5} - x^{4} + x^{2}}{-x^{2}} \] Next, we simplify this expression: \[ \frac{x^{5}}{-x^{2}} + \frac{-x^{4}}{-x^{2}} + \frac{x^{2}}{-x^{2}} = -x^{3} + x^{2} - 1 \] Thus, the expression equivalent to \( \left(\frac{u}{v}\right)(x) \) is: \[ -x^{3} + x^{2} - 1 \] Among the provided options, the correct answer is: \(-x^{3} + x^{2} - 1\)