The given expression is:

$\frac{\frac{1}{7 - 2x}}{\frac{2x^2 - x}{7x - 2x^2}}$

To simplify this expression, we can multiply by the reciprocal of the denominator.

Step 1: Identify the Reciprocal of the Denominator

The denominator of the larger fraction is $\frac{2x^2 - x}{7x - 2x^2}$. The reciprocal of this expression is:

$\frac{7x - 2x^2}{2x^2 - x}$

Step 2: Multiply the Numerator by the Reciprocal of the Denominator

Now, we multiply the numerator $\frac{1}{7 - 2x}$ by the reciprocal of the denominator:

$\frac{1}{7 - 2x} \times \frac{7x - 2x^2}{2x^2 - x}$

Step 3: Simplify the Resulting Expression

The resulting expression is:

$\frac{7x - 2x^2}{(7 - 2x)(2x^2 - x)}$

This is the simplified form of the given expression.

Final Answer:

$\frac{7x - 2x^2}{(7 - 2x)(2x^2 - x)}$

Would you like any additional details or explanations?

Here are 5 related questions to expand on this topic:

1. How do you simplify complex rational expressions involving polynomials?
2. What are the steps to factor polynomials in a rational expression?
3. How do you identify and cancel common factors in rational expressions?
4. What are the conditions under which a rational expression is undefined?
5. How does multiplying by the reciprocal help in dividing rational expressions?

Tip: When simplifying rational expressions, always check if the expressions can be factored, and if there are any common factors that can be canceled out.