The given expression is:

$\frac{-\frac{1}{2}x(1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}} x}{x^2}$

Let's simplify this step by step:

1. Factorize and group terms: $\frac{-\frac{1}{2}x(1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}}x}{x^2}$

   We can factor out $x$ from both terms in the numerator: $= \frac{x \left( -\frac{1}{2}(1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}} \right)}{x^2}$

2. Cancel $x$ in numerator and denominator: $= \frac{ -\frac{1}{2}(1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}}}{x}$

3. Simplify further: Now we have two terms involving powers of $1 - x$: $= \frac{ (1 - x)^{\frac{1}{2}} - \frac{1}{2}(1 - x)^{-\frac{1}{2}} }{x}$

Final simplified expression:

$= \frac{ (1 - x)^{\frac{1}{2}} - \frac{1}{2}(1 - x)^{-\frac{1}{2}} }{x}$

This is the simplified version of the given expression.

Would you like further details on any step? Here are some related questions to explore:

1. How do you simplify expressions involving fractional exponents?
2. What are the rules for factoring out common terms in algebraic expressions?
3. How can we handle terms involving negative exponents?
4. What is the process for cancelling terms in a fraction?
5. How does simplifying the numerator affect the overall expression?

Tip: When simplifying expressions with exponents, be mindful of how negative and fractional exponents interact with the base terms.