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9\left(x^{2}-2xy+y^{2}\right)-12\left(y^{2}-x^{2}\right)+\frac{1}{2}\times 4\left(x+y\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.
9x^{2}-18xy+9y^{2}-12\left(y^{2}-x^{2}\right)+\frac{1}{2}\times 4\left(x+y\right)^{2}
Use the distributive property to multiply 9 by x^{2}-2xy+y^{2}.
9x^{2}-18xy+9y^{2}-12y^{2}+12x^{2}+\frac{1}{2}\times 4\left(x+y\right)^{2}
Use the distributive property to multiply -12 by y^{2}-x^{2}.
9x^{2}-18xy-3y^{2}+12x^{2}+\frac{1}{2}\times 4\left(x+y\right)^{2}
Combine 9y^{2} and -12y^{2} to get -3y^{2}.
21x^{2}-18xy-3y^{2}+\frac{1}{2}\times 4\left(x+y\right)^{2}
Combine 9x^{2} and 12x^{2} to get 21x^{2}.
21x^{2}-18xy-3y^{2}+2\left(x+y\right)^{2}
Multiply \frac{1}{2} and 4 to get 2.
21x^{2}-18xy-3y^{2}+2\left(x^{2}+2xy+y^{2}\right)
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.
21x^{2}-18xy-3y^{2}+2x^{2}+4xy+2y^{2}
Use the distributive property to multiply 2 by x^{2}+2xy+y^{2}.
23x^{2}-18xy-3y^{2}+4xy+2y^{2}
Combine 21x^{2} and 2x^{2} to get 23x^{2}.
23x^{2}-14xy-3y^{2}+2y^{2}
Combine -18xy and 4xy to get -14xy.
23x^{2}-14xy-y^{2}
Combine -3y^{2} and 2y^{2} to get -y^{2}.
9\left(x^{2}-2xy+y^{2}\right)-12\left(y^{2}-x^{2}\right)+\frac{1}{2}\times 4\left(x+y\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.
9x^{2}-18xy+9y^{2}-12\left(y^{2}-x^{2}\right)+\frac{1}{2}\times 4\left(x+y\right)^{2}
Use the distributive property to multiply 9 by x^{2}-2xy+y^{2}.
9x^{2}-18xy+9y^{2}-12y^{2}+12x^{2}+\frac{1}{2}\times 4\left(x+y\right)^{2}
Use the distributive property to multiply -12 by y^{2}-x^{2}.
9x^{2}-18xy-3y^{2}+12x^{2}+\frac{1}{2}\times 4\left(x+y\right)^{2}
Combine 9y^{2} and -12y^{2} to get -3y^{2}.
21x^{2}-18xy-3y^{2}+\frac{1}{2}\times 4\left(x+y\right)^{2}
Combine 9x^{2} and 12x^{2} to get 21x^{2}.
21x^{2}-18xy-3y^{2}+2\left(x+y\right)^{2}
Multiply \frac{1}{2} and 4 to get 2.
21x^{2}-18xy-3y^{2}+2\left(x^{2}+2xy+y^{2}\right)
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.
21x^{2}-18xy-3y^{2}+2x^{2}+4xy+2y^{2}
Use the distributive property to multiply 2 by x^{2}+2xy+y^{2}.
23x^{2}-18xy-3y^{2}+4xy+2y^{2}
Combine 21x^{2} and 2x^{2} to get 23x^{2}.
23x^{2}-14xy-3y^{2}+2y^{2}
Combine -18xy and 4xy to get -14xy.
23x^{2}-14xy-y^{2}
Combine -3y^{2} and 2y^{2} to get -y^{2}.