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1)
(a+x)³-a*(a+x)²-x²*(2a+x)-a²x=(a+x)²(a+x-a)+2ax²-a²x=
=(a²+2ax+x²)*x-2ax²-x³-a²x=a²x+2ax²+x³-2ax²-x³-a²x=0.
2)
(a-1)³+3*(a-1)²+3*(a-1)+1-a³=(a-1)²*(a-1+3)+3a-3+1-a³=
=(a²-2a+1)*(a+2)+3a+-2-a³=a³-2a²+a+2a²-4a+2+3a-2-a³=0.
3)
(x³+y³)²-(x²+y²)³+3x²y²*(x+y)²-8x³y³=
=x⁶+2x³y³+y⁶-x⁶-3x⁴y²-3x²y⁴-y⁶+3x²y²*(x²+2xy+y²)-8x³y³=
=2x³y³-3x⁴y²-3x²y⁴+3x⁴y²+6x³y³+3x²y⁴-8x³y³=0.
4)
(m-3n)³-(2m-3n)(3mn+(m-3n)²)+m³=
=m³-9m²n+27mn²-27n³-(2m-3n)*(3mn+m²-6mn+9n²)+m³=
=m³-9m²n+27mn²-27n³-                                                                                                    -(6m²n+2m³-12m²n+18mn²-9mn²-3m²n+18mn²-27n³)+m³=
=2m³-9m²n+27mn²-27n³-(-9m²n+2m³+27mn²-27n³)=
=2m³-9m²n+27mn²-27n³+9m²n-2m³-27mn²+27n³=0.

1)
(a+x)³-a*(a+x)²-x²*(2a+x)-a²x=(a+x)²(a+x-a)+2ax²-a²x=
=(a²+2ax+x²)*x-2ax²-x³-a²x=a²x+2ax²+x³-2ax²-x³-a²x=0.
2)
(a-1)³+3*(a-1)²+3*(a-1)+1-a³=(a-1)²*(a-1+3)+3a-3+1-a³=
=(a²-2a+1)*(a+2)+3a+-2-a³=a³-2a²+a+2a²-4a+2+3a-2-a³=0.
3)
(x³+y³)²-(x²+y²)³+3x²y²*(x+y)²-8x³y³=
=x⁶+2x³y³+y⁶-x⁶-3x⁴y²-3x²y⁴-y⁶+3x²y²*(x²+2xy+y²)-8x³y³=
=2x³y³-3x⁴y²-3x²y⁴+3x⁴y²+6x³y³+3x²y⁴-8x³y³=0.
4)
(m-3n)³-(2m-3n)(3mn+(m-3n)²)+m³=
=m³-9m²n+27mn²-27n³-(2m-3n)*(3mn+m²-6mn+9n²)+m³=
=m³-9m²n+27mn²-27n³-                                                                                                    -(6m²n+2m³-12m²n+18mn²-9mn²-3m²n+18mn²-27n³)+m³=
=2m³-9m²n+27mn²-27n³-(-9m²n+2m³+27mn²-27n³)=
=2m³-9m²n+27mn²-27n³+9m²n-2m³-27mn²+27n³=0.