Given the inequalities $x\(<0<y<1\)$, we need to determine which of the provided options correctly orders the following expressions from least to greatest:

$$
\(x^2, \quad(x y)^2, \quad x^2 y, \quad\left(\frac{x}{y}\right)^2, \quad \frac{x^2}{y}\)
$$


Step 1: Understand the Given Conditions
- $\(x<0\)$ : $x$ is negative.
- $\(0<y<1\)$ : $y$ is a positive fraction.

Step 2: Analyze Each Expression
Let's evaluate each expression under the given conditions.

1. $\(x^2\)$ :
- Since $x$ is negative, $\(x^2\)$ is positive.
$\(\circ x^2=(-|x|)^2=|x|^2\)$.

2. $(x y)^2$ :
$\(\circ x\)$ is negative, $y$ is positive $\rightarrow x y$ is negative.
- Squaring a negative number gives a positive result: $\((x y)^2=(|x| y)^2=x^2 y^2\)$.
- Since $\(0<y<1, y^2<y\)$, so $\((x y)^2=x^2 y^2<x^2 y\)$.

3. $\(x^2 y\)$ :

$\(\circ x^2\)$ is positive, $y$ is a positive fraction $\(\rightarrow x^2 y\)$ is positive but less than $\(x^2\)$ (since $\(y<1\)$ ).

4. $\(\left(\frac{x}{y}\right)^2\)$ :

- $\(\frac{x}{y}\)$ is negative (negative divided by positive).

- Squaring gives a positive result: $\(\left(\frac{x}{y}\right)^2=\frac{x^2}{y^2}\)$.

$\(\circ\)$ Since $\(y<1, y^2<y\)$, so $\(\frac{x^2}{y^2}>\frac{x^2}{y}\)$.

5. $\(\frac{x^2}{y}\)$ :

$\(\circ x^2\)$ is positive, $y$ is a positive fraction $\(\rightarrow \frac{x^2}{y}\)$ is positive and greater than $\(x^2\)$ (since $\(y<1\)$ ).


Step 3: Compare the Expressions

Now, let's compare the expressions based on the above analysis:

1. $\((x y)^2$ vs. $x^2 y\)$ :

- $\((x y)^2=x^2 y^2\)$.
- Since $\(y<1, y^2<y\)$, so $\(x^2 y^2<x^2 y\)$.

2. $\(x^2 y\)$ vs. $\(x^2\)$ :

- $\(y<1\)$, so $\(x^2 y<x^2\)$.

3. $\(x^2\)$ vs. $\(\frac{x^2}{y}\)$ :

$\(\circ y<1\)$, so $\(\frac{x^2}{y}>x^2\)$.

4. $\(\frac{x^2}{y}\)$ vs. $\(\left(\frac{x}{y}\right)^2\)$ :

- $\(\left(\frac{x}{y}\right)^2=\frac{x^2}{y^2}\)$.
$\(\circ\)$ Since $\(y<1, y^2<y\)$, so $\(\frac{x^2}{y^2}>\frac{x^2}{y}\)$.

Step 4: Establish the Order
From the comparisons, the order from least to greatest is:

$$
\((x y)^2<x^2 y<x^2<\frac{x^2}{y}<\left(\frac{x}{y}\right)^2\)
$$


Step 5: Match with the Given Options
Comparing with the options:
- (A): Incorrect order.
- (B): Incorrect order.
- (C): Incorrect order.
- (D): Incorrect order.
- (E): Matches our derived order.

Conclusion

The correct option is (E).