Questions: Which of the tollowing is the quotient of the rational expressions shown below? Make sure your answer is in reduced form. (3x-6)/(x^3) ÷ (x-2)/(2x-1) A. (4x-8)/(x^3+2x-1) B. (3x^2-12x+12)/(2x^4-x^3) C. (6x^2-15x+6)/(x^4-2x^3) D. (5x-7)/(x^3+x-2) E. (6x-3)/(x^3)

Determine the quotient of the rational expressions \( \frac{3x - 6}{x^3} \div \frac{x - 2}{2x - 1} \).

Rewrite the division as multiplication.

The expression can be rewritten as \( \frac{3x - 6}{x^3} \times \frac{2x - 1}{x - 2} \).

Factor the numerator and denominator.

The numerator \( 3x - 6 \) factors to \( 3(x - 2) \), and the denominator \( x^3 \) remains as is. Thus, we have: \[ \frac{3(x - 2)}{x^3} \times \frac{2x - 1}{x - 2} \]

Cancel common factors.

The \( (x - 2) \) in the numerator and denominator cancels out, resulting in: \[ \frac{3(2x - 1)}{x^3} \]

The simplified result is \( \boxed{\frac{3(2x - 1)}{x^3}} \).

Identify the final simplified expression.

Present the simplified expression.

The final simplified expression is \( \frac{3(2x - 1)}{x^3} \).

Compare with the provided options.

The expression \( \frac{3(2x - 1)}{x^3} \) does not match any of the provided options directly, but can be expressed as \( \frac{6x - 3}{x^3} \) by distributing the 3 in the numerator.

The closest match is \( \boxed{\frac{6x - 3}{x^3}} \).

The simplified result of the quotient is \( \boxed{\frac{3(2x - 1)}{x^3}} \).
The closest match to the simplified expression is \( \boxed{\frac{6x - 3}{x^3}} \).