Verified answer

81x^4 + 50x^2y^2 + 625y^4

= 81x^4 + 450x^2y^2 + 625y^4 – 400x^2y^2

= (9x^2 + 25y^2)^2 – (20xy)2^

= (9x^2 + 25y^2 + 20xy)(9x^2 + 25y^2 – 20xy)

= (9x^2 + 20xy + 25y^2)(9x^2 – 20xy + 25y^2)

further factors if required

(9x^2 + 20xy + 25y^2)

= (9x^2 + 30xy + 25y^2) – 10xy

= (3x + 5y) – (√10xy)^2

= (3x + 5y + √10xy)(3x + 5y – (√10xy)

------

50^2 - 4*81*625 = -200000

There will not be any real factors,

and the imaginary parts of the complex factors will not have rational coefficients.

So just say "unfactorable".