${\left(x-2y\right)}^3{\left(x+2y\right)}^3+{(x^3+8y^3)}^2-4x^2{y}^2\left(-3x^2+12y^2+4\mathrm{xy}\right)=$

$={(x^2-4y^2)}^3+x^6+16x^3{y}^3+64y^6+12x^4{y}^2-48x^2{y}^4-16x^3{y}^3=$

$=x^6{+48x^2{y}^4-12x^4{y}^2-64y^6+x}^6+16x^3{y}^3+64y^6+12x^4{y}^2-48x^2{y}^4-16x^3{y}^3=$

$=\left(1+1\right)x^6+\left(-64+64\right)y^6+\left(48-48\right)x^2{y}^4+\left(-12+12\right)x^4{y}^2+\left(16-16\right)x^3{y}^3=2x^6$