That's the sum of two squares. But we can only factor the difference of two squares, so we see if it is possible to change it to the difference of two squares by adding and subtracting a square, like this

x⁴ + ___ + 4y⁴ - ___

We ask ourselves this question: What term would have to be placed in those blanks so that the first three terms would factor as the square of the sum of their square roots, x² and 2y²?

To find out we multiply out the square of the sum of their square roots: (x² + 2y²)² = x⁴ + 4x²y² + 4y⁴

We see that the term that must be placed in the two blanks would be 4x²y², which does happen to be a square.

So we place 4x²y² in the two blanks: x⁴ + 4x²y² + 4y⁴ - 4x²y²

Then we factor the first three terms: (x²+2y²)² - 4x²y² and that is the difference of two squares, and factors as

[(x²+2y²) - 2xy][(x+2y²) + 2xy]

Removing the parentheses inside the brackets: [x² + 2y² - 2xy][x² + 2y² + 2xy]

Then we can change the brackets to parentheses and rearrange the trinomials in descending order of x and ascending order of y:

(x² - 2xy + 2y²)(x² + 2xy + 2y²)