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".
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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".