A=(x+y)\(^3\)+z\(^3\)+3(x+y)z(x+y+z)-x\(^3\)-y\(^3\)-z\(^3\)                                                                                                    

   =x\(^3\)+y\(^3\)+3xy(x+y)+3(x+y)z(x+y+z)-x\(^3\)-y\(^3\)

   =3(x+y)(xy+zx+zy+z\(^2\))

    =3(x+y)\([\)x(y+z)+z(y+z)\(]\)

    =3(x+y)(y+z)(x+z)

  (bạn chỉ cần dùng hằng đẳng thức là ok ngay)

Đặt \(x^2+x+1=t\)

\(\left(x^2+x+1\right)\left(x^2+x+2\right)-12=t\left(t+1\right)-12=t^2+t-12=\left(t^2+t+\dfrac{1}{4}\right)-\dfrac{49}{4}=\left(t+\dfrac{1}{2}\right)^2-\left(\dfrac{7}{2}\right)^2=\left(t+\dfrac{1}{2}-\dfrac{7}{2}\right)\left(t+\dfrac{1}{2}+\dfrac{7}{2}\right)=\left(t-3\right)\left(t+4\right)=\left(x^2+x-2\right)\left(x^2+x+5\right)\)

\(\left(x^2+x+1\right)\left(x^2+x+2\right)-12\)

= \(\left(x^2+x+1\right)\left[\left(x^2+x+1\right)+1\right]-12\)

= \(\left(x^2+x+1\right)^2\left(x^2+x+1\right)-12\)

= \(\left(x^2+x+1\right)\left(x^2+x+1\right)-3\left(x^2+x+1\right)+4\left(x^2+x+1\right)-4.3\)

= \(\left(x^2+x+1\right)\left(x^2+x-2\right)+4\left(x^2+x-2\right)\)

= \(\left(x^2+x+5\right)\left(x^2+x-2\right)\)

Đây nha c ơi

Làm như vầy là sai hướng rồi.

Tham khảo :

\(\left(x+y+z\right)^3-x^3-y^3-z^3\)

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

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

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

\(=\left(y+z\right)\left[3x^2+3xy+3yz+3xz\right]\)

\(=3\left(y+z\right)\left[\left(x^2+xy\right)+\left(yz+xz\right)\right]\)

\(=3\left(y+z\right)\left[x\left(x+y\right)+z\left(x+y\right)\right]\)

\(=3\left(x+y\right)\left(y+z\right)\left(x+z\right)\)

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

\(=-y^3-xy^2+x^2y+x^3-z^3-yz^2+y^2z+y^3-x^3-zx^2+z^2x+z^3\)

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

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

Đặt \(x+y-z=a;x-y+z=b;y+z-x=c\)

Ta có:\(A=\left(a+b+c\right)^3-a^3-b^3-c^3\)

\(A=\left[\left(a+b\right)+c\right]^3-a^3-b^3-c^3\)

\(A=\left(a+b\right)^3+3\left(a+b\right)\cdot c\cdot\left(a+b+c\right)+c^3-a^3-b^3-c^3\)

\(A=a^3+b^3+3ab\left(a+b\right)+3\left(a+b\right)c\left(a+b+c\right)+c^3-a^3-b^3-c^3\)

\(A=3\left(a+b\right)\left(ab+ac+bc+c^2\right)\)

\(A=3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

Hay \(A=3\cdot2x\cdot2y\cdot2z\)

\(A=24xyz\)