APPROACH #1: Algebra
Given: $\frac{2z^2(z-1)+z-z^2}{z(z-1)}$

Rearrange the last two terms in the numerator: $\frac{2z^2(z-1)-z^2+z}{z(z-1)}$

Factor out $-z$ from the last two terms of numerator: $\frac{2z^2(z-1)-z(z-1)}{z(z-1)}$

Rewrite the numerator as follows: $\frac{(2z^2-z)(z-1)}{z(z-1)}$

Simplify: $\frac{2z^2-z}{z}$

Finally, we can divide numerator and denominator by $z$ to get: $2z-1$

Answer: D


APPROACH #2: Test a value of z
Since we are looking for an expression that's equivalent to $\frac{2z^2(z-1)+z-z^2}{z(z-1)}$, let's first evaluate this expression for a certain value of $z$, and then look for an answer choice that has the same value for that value of $z$

Let's see what happens when we plug $z = 2$ into the given expression:
$\frac{2z^2(z-1)+z-z^2}{z(z-1)} = \frac{2(2^2)(2-1)+2-2^2}{2(2-1)}$ $= \frac{8+2-4}{2}$ $= \frac{6}{2} = 3$

So, the given expression evaluates to equal $3$ when $z = 2$.

We can now plug $z = 2$ into each answer choice to see which one evaluates to $3$

(A) $2(2^2) − 2 =6$. ELIMINATE
(B) $2(2) + 1=5$. ELIMINATE
(C) $2(2)=4$. ELIMINATE
(D) $2(2) − 1=3$. KEEP
(E) $2 − 2=0$. ELIMINATE

Answer: D