Factor the polynomial $7x^3-14x^{2}y+2x{y}^2-4y^3$ by grouping.

A) $(7x-y)\left(x-4y^3\right)$
B) $(7x-y)\left(x+4y^3\right)$
C) $(x-2y)\left(7x^2-2y^2\right)$
D) $(x-2y)\left(7x^2+2y^2\right)$
E) $(x-14y)\left(7x^2-4y^2\right)$

The first step in grouping is to factor out the greatest common factor from the entire polynomial:

$$7x^3-14x^{2}y+2x{y}^2-4y^3=7x(x^2-2x{y}^2)+2y^2(-2x+2y)$$

Now we can group the two terms on the left that have a common factor of $x$ and factor out that common factor, and group the two terms on the right that have a common factor of $2y^2$ and factor out that common factor:

$$7x(x^2-2x{y}^2)+2y^2(-2x+2y)=x(7x^2-14x{y}^2)+2y^2(-2x+2y)$$

Now we can factor out an additional factor of $x-2y$ from each of the terms in the expression:

$$x(7x^2-14x{y}^2)+2y^2(-2x+2y)=(x-2y)(7x^2+2y^2)$$

So the polynomial factors as $(x-2y)(7x^2+2y^2)$ , which corresponds to choice D.