Key idea. Create a square by adding and subtracting the missing middle term.

1. We know \((x^2+2y^2)^2=x^4+4x^2y^2+4y^4\). Therefore

   \[x^4+4y^4=(x^2+2y^2)^2-4x^2y^2.\]

2. The subtracted term is also a square: \(4x^2y^2=(2xy)^2\).

3. Apply the difference-of-squares identity:

   \[x^4+4y^4=(x^2-2xy+2y^2)(x^2+2xy+2y^2).\]

Remember. A sum can sometimes be turned into a difference of squares. This identity is often called Sophie Germain’s identity.