Factor:

x^8 + 2x^4 y^4 + 9y^8

To factor the given expression, x^8 + 2x^4y^4 + 9y^8, we can use the formula for sum of cubes:

a^3 + b^3 = (a + b)(a^2 - ab + b^2).

Notice that the given expression, x^8 + 2x^4y^4 + 9y^8, can be rearranged as:

(x^8 + 2x^4y^4 + y^8) + 8y^8
= (x^8 + 2x^4y^4 + y^8) + (3y^4)^2
= (x^4 + y^4)^2 + (3y^4)^2.

Now, we can rewrite it as the sum of two cubes:

(a^3 + b^3) = (a + b)(a^2 - ab + b^2),

where a = (x^4), and b = (3y^4).

Substituting these values, we have:

(x^8 + 2x^4y^4 + 9y^8) = [(x^4) + (3y^4)][(x^4)^2 - (x^4)(3y^4) + (3y^4)^2].

Simplifying further, we get:

(x^8 + 2x^4y^4 + 9y^8) = (x^4 + 3y^4)(x^4^2 - 3x^4y^4 + 9y^8).

Therefore, the factored form of the expression x^8 + 2x^4y^4 + 9y^8 is (x^4 + 3y^4)(x^4^2 - 3x^4y^4 + 9y^8).