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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Phức tạp. Cs cách nào nhanh kkk?

xy(x+y)+yz(y+z)+xz(x+z)+2xyz
= xy(x + y) + yz(y + z) + xyz + xz(x + z) + xyz
= xy(x + y) + yz(y + z + x) + xz(x + z + y)
= xy(x + y) + z(x + y + z)(y + x)
= (x + y)(xy + zx + zy + z²)
= (x + y)[x(y + z) + z(y + z)]
= (x + y)(y + z)(z + x)

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

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

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

\(\Leftrightarrow y\left(x+y+z\right)\left(x+z\right)+xz\left(x+z\right)\)
\(\Leftrightarrow\left(x+z\right)\left(y\left(z+x\right)+zx\right)\)

\(\Leftrightarrow\left(x+z\right)\left(y+z\right)\left(x+y\right)\)

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

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

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

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

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

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

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

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

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

\(=\left(y+z\right)\left(x+y\right)\left(x+2\right)\)

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

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

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

\(=\left(y-z\right)\left[x^2+yz-x\left(y+z\right)\right]\)

\(=\left(y-z\right)\left[x\left(x-y\right)-z\left(x-y\right)\right]\)

\(=\left(y-z\right)\left[\left(x-z\right)\left(x-y\right)\right]\)

=[xy(x+y)+xyz]+[yz(y+z)+xyz]+xz(x+z)

=xy(x+y+z)+yz(x+y+z)+xz(x+z)

=y(x+y+z)(x+z)+xz(x+z)

=(x+z)(xy+y²+yz+xz)

=(x+z)(x+y)(y+z)

Thanks

hic hic khó quá đi