What is the completely factored form of xy^3 - x^3y?

1. Identify the expression: The expression to be factored is $xy^3 - x^3y$.
2. Find the greatest common factor (GCF): Observe that both terms have common factors of $x$ and $y$. The lowest power of $x$ is $x^1$ and the lowest power of $y$ is $y^1$. Therefore, the GCF is $xy$.
3. Factor out the GCF: Divide each term by the GCF:
$xy^3 / xy = y^2$
$x^3y / xy = x^2$
So, the expression becomes $xy(y^2 - x^2)$.
4. Factor the difference of squares: The expression inside the parentheses, $y^2 - x^2$, is a difference of squares, which can be factored as $(a-b)(a+b)$. In this case, $a=y$ and $b=x$. So, $y^2 - x^2 = (y-x)(y+x)$.
5. Combine the factors: Substitute the factored difference of squares back into the expression:
$xy(y-x)(y+x)$.
6. Check for further factorization: The factors $(y-x)$ and $(y+x)$ are linear and cannot be factored further. The factor $xy$ is also factored completely. Therefore, the completely factored form is $xy(y-x)(y+x)$.

Alternatively, we can write $y^2 - x^2$ as $-(x^2 - y^2)$, which factors as $-(x-y)(x+y)$.
Then the expression becomes $xy(-(x-y)(x+y)) = -xy(x-y)(x+y)$.
However, looking at the options provided, the form $xy(y-x)(y+x)$ is present. Let's re-examine the options and the factoring process.

Let's consider the option $xy(x-y)(x^2+xy+y^2)$. Expanding this:
$xy(x^3+x^2y+xy^2 - yx^2 - xy^2 - y^3) = xy(x^3 - y^3)$. This is not the original expression.

Let's consider the option $xy(y-x)(y^2+xy+x^2)$. Expanding this:
$xy(y(y^2+xy+x^2) - x(y^2+xy+x^2))$
$= xy(y^3 + xy^2 + x^2y - xy^2 - x^2y - x^3)$
$= xy(y^3 - x^3)$. This is not the original expression.

Let's go back to step 4. We factored $y^2 - x^2$ as $(y-x)(y+x)$.
So the expression is $xy(y-x)(y+x)$.

Let's re-examine the original expression and the options. The original expression is $xy^3 - x^3y$.
Factoring out $xy$: $xy(y^2 - x^2)$.
Factoring the difference of squares: $xy(y-x)(y+x)$.

Let's check the provided options again.
Option 1: $xy(x-y)(x^2+xy+y^2)$. This is $xy(x^3-y^3)$. Incorrect.
Option 2: $xy(y-x)(y-x)$. This is $xy(y-x)^2$. Incorrect.
Option 3: $xy(x-y)(y^2+xy+x^2)$. This is $xy(x-y)(x^2+xy+y^2)$. This is $xy(x^3-y^3)$. Incorrect.
Option 4: $xy(y+x)(y-x)$. This is $xy(y^2-x^2)$. This is the correct factorization of $xy^3 - x^3y$.

Let's re-evaluate the problem and options. The problem asks for the completely factored form of $xy^3 - x^3y$.
We factored out $xy$ to get $xy(y^2 - x^2)$.
The difference of squares $y^2 - x^2$ factors into $(y-x)(y+x)$.
So the completely factored form is $xy(y-x)(y+x)$.

Looking at the provided options, the third option is $xy(x-y)(y^2+xy+x^2)$. This expands to $xy(x^3 - y^3)$. This is incorrect.

Let's assume there might be a typo in the options or the problem statement. However, based on standard algebraic factoring, $xy^3 - x^3y = xy(y^2 - x^2) = xy(y-x)(y+x)$.

Let's consider the possibility that the question intended to ask for the factorization of $x^3y - xy^3$. In that case, $x^3y - xy^3 = xy(x^2 - y^2) = xy(x-y)(x+y)$. This is also not among the options in that exact form.

Let's re-examine the provided options and the original expression.
Original expression: $xy^3 - x^3y$.
Factor out $xy$: $xy(y^2 - x^2)$.
Factor difference of squares: $xy(y-x)(y+x)$.

Let's check the options again.
Option 1: $xy(x-y)(x^2+xy+y^2) = xy(x^3-y^3)$. Incorrect.
Option 2: $xy(y-x)(y-x) = xy(y-x)^2$. Incorrect.
Option 3: $xy(x-y)(y^2+xy+x^2) = xy(x^3-y^3)$. Incorrect.
Option 4: $xy(y+x)(y-x)$. This is $xy(y^2-x^2)$. This matches our factorization.

Therefore, the correct option is $xy(y+x)(y-x)$.

Let's verify the expansion of $xy(y+x)(y-x)$:
$xy(y^2 - x^2) = xy^3 - x^3y$. This matches the original expression.

Final check of the steps:
1. Factor out GCF: $xy^3 - x^3y = xy(y^2 - x^2)$.
2. Factor difference of squares: $y^2 - x^2 = (y-x)(y+x)$.
3. Combine: $xy(y-x)(y+x)$.
This can also be written as $xy(y+x)(y-x)$ due to the commutative property of multiplication.

Comparing with the options, the fourth option is $xy(y+x)(y-x)$.