Bài 3: 

\(B=x^4-4x^3-2x^2+12x+9\)

\(=x^4-3x^3-x^3+3x^2-5x^2+15x-3x+9\)

\(=\left(x-3\right)\left(x^3-x^2-5x-3\right)\)

\(=\left(x-3\right)\left(x^3-3x^2+2x^2-6x+x-3\right)\)

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

\(=\left(x^2-2x-3\right)^2\)

Bài 3: 

\(B=x^4-4x^3-2x^2+12x+9=\left(x^4+x^3\right)-\left(5x^3+5x^2\right)+\left(3x^2+3x\right)+\left(9x+9\right)=\left(x^3-5x^2+3x+9\right)\left(x+1\right)=\left[\left(x^3+x^2\right)-\left(6x^2+6x\right)+\left(9x+9\right)\right]\left(x+1\right)=\left(x^2-6x+9\right)\left(x+1\right)^2=\left(x-3\right)^2\left(x+1\right)^2=\left[\left(x-3\right)\left(x+1\right)\right]^2\)

Bạn tính toán cuối cùng nó ra M=(yz+2xz+2xy+2x²)²

x;y;z là các số tự nhiên => M là số chính phương

Cảm ơn

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\(M=4x\left(x+y+z\right)\left(x^2+xz+yx+yz\right)+\left(yz\right)^2\)

\(M=4\left(x^2+xy+zx\right)\left(x^2+yz+zx+xy\right)+\left(yz\right)^2\)

\(M=4\left(x^2+xy+zx\right)\left\{\left(x^2+yz+zx\right)+xy\right\}+\left(yz^2\right)\)

\(M=4\left(x^2+xy+zx\right)^2+4\left(x^2+yz+zx\right)\left(yz\right)+\left(yz\right)^2\) ( hằng đẳng thức )

\(M=\left\{2\left(x^2+xy+zx\right)\right\}^2+2.2\left(x^2+xy+zx\right)\left(yz\right)+\left(yz\right)^2\)

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

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

\(M=4x\left(x+y\right)\left(x+y+z\right)\left(x+z\right)+y^2z^2\)

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

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

Đặt \(2x^2+2xy+2xz+yz=a\)

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

\(=a^2-y^2z^2+y^2z^2\)

\(=a^2\)

Mà \(x;y;z\in N\Rightarrow a\in N\)

=> M là số chính phương

\(M=4\left(x^2+xy+xz\right)\left(x^2+xy+yz+zx\right)+y^2z^2\)

Đặt \(a=x^2+xy+xz\)

\(M=4a\left(a+yz\right)+y^2z^2=4a^2+4ayz+y^2z^2=\left(2a+yz\right)^2\)

Vậy \(M=\left(2x^2+2xy+2xz+yz\right)^2\)là số chính phương

B= 4(x² + xy + xz)(x² + xy + xz + yz) + y²z²

đặt x² + xy + xz = m , ta có

B = 4m(m + yz) + y²z² = 4m² + 4myz + y²z²

B = (2m + yz)² = (2x² + 2xy + 2xz + yx)²

x,y,z la cac so nguyen thif B la 1 so chinh phuong

abcdacscas

B3 : t chỉ m r á :3
B4 : 
Ta có :
C= 4x ( x + y ) ( x + y + z ) ( y + z ) + y²x²
   = 4x ( x + y + z ) ( x + y ) ( x + z ) + y²x²
   = 4 ( x² + xy + xz ) ( x² + xy + xz + yz ) + y²x²
Đặt a = x² + xy + xz và b= yz , ta có :
  ⇒ C = 4a( a + b ) + b²
          = b² + 4ab + 4a²
          = ( b + a )²
  ⇒ C là số chính phương 
Chúc mừng m đã ghi xong bài , nhớ tick cho t nhoa bff!
            

\(C=4x\left(x+y+z\right)\left(x+y\right)\left(x+z\right)+y^2z^2=4\left(x^2+xy+xz\right)\left(x^2+xz+xy+yz\right)+y^2z^2\)

Đặt \(x^2+xy+xz=t\), ta có:

\(C=4t\left(t+yz\right)+y^2z^2=4t^2+4tyz+y^2z^2=\left(2t\right)^2+2.2t.yz+\left(yz\right)^2=\left(2t+yz\right)^2=\left(2x^2+2xy+2xz+yz\right)^2\)

=>đpcm