\(A=\left(x^2+y^2+z^2\right)\left(x+y+z\right)^2-\left(xy+yz+zx\right)^2\left(1\right)\)

Đặt \(x^2+y^2+z^2=a\)

\(xy+yz+zx=b\Rightarrow2\left(xy+yz+zx\right)=2b\)

\(\Rightarrow a+2b=\left(x+y+z\right)^2\)

Kết hợp (1) ta được : \(A=a\left(a+2b\right)+b^2\)

                                      \(=a^2+2ab+b^2\)

                                     \(=\left(a+b\right)^2\)

                                      \(=\left(x^2+y^2+z^2+xy+yz+zx\right)^2\)

       \(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2+3xyz\)

\(=\left(x^2y+xy^2+xyz\right)+\left(x^2z+xz^2+xyz\right)+\left(y^2z+yz^2+xyz\right)\)

\(=xy\left(x+y+z\right)+xz\left(x+y+z\right)+yz\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(xy+xz+yz\right)\)

nick ko hay rồi tcn còn ko hay nữa

\(2\left(xy+yz+zx\right)-x^2-y^2-z^2\)

\(2xy+2yz+2zx-x^2-y^2-z^2\)

\(-\left(x^2+y^2+z^2-2xy-2yz-2xz\right)\)

\(-\left(x+y+z\right)^2\)

= xyx + xyy - yzy + yzz - zx( z - x ) 

= y( x^2 + xy ) - y( zy + zz ) - zx( z - x ) 

= y[ ( x^2 + xy ) - ( zy + zz ) ] - zx( z - x ) 

= y( x^2 + xy - zy - zz ) - zx( z - x ) 

= y[ x( x + y ) - z( y - z ) ] - zx( z - x ) 

P/S : bí rùi . ngu phần này lắm . 

xy x dc goi la một đương f thẳng nên nó sẻ dc goi 

\(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2+2xyz.\)

\(=x^2.\left(y+z\right)+yz.\left(y+z\right)+x\left(y^2+z^3\right)+2xyz\)

\(=\left(y+z\right).\left(x^2+yz\right)+x\left(y^{^2}+z^2+2yz\right)\)

\(=\left(y+z\right).\left[x.\left(x+2\right)+y.\left(x+2\right)\right]\)

\(=\left(y+z\right).\left(x+z\right).\left(x+y\right)\)