`\color{#00EDFF}{circ} \color{#1AD5F7}{H} \color{#1AD5F7} {e} \color{#4DA6E6}{r} \color{#668EDD}{e} \color{#8077D5} {n} \color{#995FCD}{a}`

`(x+y+z)^5 - x^5 -y^5 -z^5`

`=(x+y)^5 + 5(x+y)^(4)z + 10(x+y)^(3)z^2 + 10(x+y)^(2)z^3 + 5(x+y)z^4 + z^5 - x^5 - y^5 - z^5`

`=(x+y)^5 + 5(x+y)^(4)z + 10(x+y)^(3)z^2 + 10(x+y)^(2)z^3 + 5(x+y)z^4 - (x+y)(x^4 - x^3y + x^(2)y^(2) - xy^3 + y^4)`

`= (x+y)[(x+y)^4 + 5(x+y)^(3)z + 10(x+y)^(2)z^2 + 10(x+y)z^3 + 5z^4 - x^4 + x^3y - x^2y^2 + xy^3 - y^4]`

`= (x+y)[x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4 + 5(x+y)^(3)z + 10(x+y)^(2)z^2 + 10(x+y)z^3 + 5z^4 - x^4 + x^3y - x^2y^2 + xy^3 - y^4]`

`= (x+y)[5x^3y + 5x^2y^2 + 5xy^3 + 5(x+y)^3z + 10(x+y)^(2)z^2 + 10(x+y)z^3 + 5z^4 ]`

`=5(x+y)[x^3y + x^2y^2 + xy^3 + x^3z + 3x^2yz + 3xy^2z + y^3z + 2x^2z^2 + 4xyz^2 + 2y^2z^2 + 2xz^3 + 2yz^3 + z^4]`

`=5(x+y)[(x^3y + x^3z) + (x^2y^2 + x^2yz) + (2x^2yz + 2x^2z^2) + (xy^3z + xy^2z) + (2xy^2z + 2xyz^2) + (2xyz^2 + 2xz^3) + (y^3z + y^2z^2) + (y^2z^2 + yz^3) + (yz^3 + z^4)]`

`=5(x+y)(y+z)[x^3 + x^2y + 2x^2z + xy^2 + 2xyz + 2xz^2 + y^2z + yz^2 + z^3]`

`=5(x+y)(y+z)[(x^3 + x^2z) + (x^2z + 2xz^2 + z^3) + (x^2y + 2xyz + yz^2) + (xy^2 + y^2z)]`

`=5(x+y)(y+z)(x+z)(x^2 + y^2 + z^2 + xy + yz + zx)`