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x^{2}+xy+xz-yx-y^{2}-yz-zx-zy-z^{2}
Apply the distributive property by multiplying each term of x-y-z by each term of x+y+z.
x^{2}+xz-y^{2}-yz-zx-zy-z^{2}
Combine xy and -yx to get 0.
x^{2}-y^{2}-yz-zy-z^{2}
Combine xz and -zx to get 0.
x^{2}-y^{2}-2yz-z^{2}
Combine -yz and -zy to get -2yz.
x^{2}+xy+xz-yx-y^{2}-yz-zx-zy-z^{2}
Apply the distributive property by multiplying each term of x-y-z by each term of x+y+z.
x^{2}+xz-y^{2}-yz-zx-zy-z^{2}
Combine xy and -yx to get 0.
x^{2}-y^{2}-yz-zy-z^{2}
Combine xz and -zx to get 0.
x^{2}-y^{2}-2yz-z^{2}
Combine -yz and -zy to get -2yz.