Divide. \[ \frac{5 x-x^{2}}{x^{3}-125} \div \frac{x}{x^{2}+5 x+25} \]

To solve the expression \[ \frac{5x - x^2}{x^3 - 125} \div \frac{x}{x^2 + 5x + 25}, \] we can first rewrite the division as multiplication by the reciprocal. This gives us: \[ \frac{5x - x^2}{x^3 - 125} \cdot \frac{x^2 + 5x + 25}{x}. \] Next, let's simplify each component. 1. **Factor the numerators and denominators**: - The numerator \( 5x - x^2 \) can be factored as \( -x(x - 5) \). - The denominator \( x^3 - 125 \) is a difference of cubes and can be factored as \( (x - 5)(x^2 + 5x + 25) \). Now, substitute these factorizations in: \[ \frac{-x(x - 5)}{(x - 5)(x^2 + 5x + 25)} \cdot \frac{x^2 + 5x + 25}{x}. \] 2. **Cancel common terms**: - The \( (x - 5) \) in the numerator can be canceled with the \( (x - 5) \) in the denominator. - The \( x \) in the numerator from \( -x \) can be canceled with \( x \) in the \(\frac{x^2 + 5x + 25}{x}\). This gives us: \[ \frac{-1}{1} \cdot (x^2 + 5x + 25) = -(x^2 + 5x + 25). \] Thus, the final simplified expression is: \[ -x^2 - 5x - 25. \]