How to simplify 5x^3 y+20x^2 y^2+20xy^3 over 5xy

The answer is (x+2y)^2 but i don't understand why

5x^3y+20x^2 y^2+20xy^3 =
5xy*(x^2+4xy+4y^2)=
5xy*(x+2y)(x+2y)=
5xy*(x+2y)^2
If you divide by 5xy that leaves (x+2y)^2

5x^3 y+20x^2 y^2+20xy^3 / 5xy

If looked at with an actual fraction we can see that the 5xy in the denominator will cancel with things in the numerator until we get:

x^2 + 4xy + 4y^2

This can be factored to

(x+2y)(x+2y) = (x+2y)^2

idk