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\(=\left(x+y\right)\left(x^2-xy+y^2\right)+2x\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2+2x\right)\)
c: \(x^2-4+3\left(x-2\right)^2\)
\(=\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(3x-6\right)\)
\(=\left(x-2\right)\left(x+2+3x-6\right)\)
\(=\left(4x-4\right)\left(x-2\right)\)
\(=4\left(x-1\right)\left(x-2\right)\)
\(\dfrac{1}{4}x^2+2xy+4y^2=\left(\dfrac{1}{2}x+2y\right)^2\)
\(x^2+2xy-8y^2+2xz+14yz-3z^2\)
\(=\left(x^2+y^2+z^2+2xy+2xz+2yz\right)+\left(-9x^2+12yz-4x^2\right)\)
\(=\left(x+y+z\right)^2-\left[\left(3x\right)^2-2.3x.2y+\left(2x\right)^2\right]\).
\(=\left(x+y+z\right)^2-\left(3y-2x\right)^2\)
\(=\left(x+y+z-3y+2x\right)\left(x+y+z+3y-2x\right)\)
\(x^4-x^2+2x+2\)
\(=x^4-2x^3+2x^2+2x^3-4x^2+4x+x^2-2x+2\)
\(=\left(x^4-2x^3+2x^2\right)+\left(2x^3-4x^2+4x\right)+\left(x^2-2x+2\right)\)
\(=x^2\left(x^2-2x+2\right)+2x\left(x^2-2x+2\right)+\left(x^2-2x+2\right)\)
\(=\left(x^2-2x+2\right)\left(x^2+2x+1\right)\)
\(=\left(x^2-2x+2\right)\left(x+1\right)^2\)
x2 + 2xy - 8y2 + 2xz + 14yz - 3z2
= ( x2 + y2 +z2 + 2xy + 2yz ) + ( -9x2 + 12yz - 4x2 )
= ( x + y +z )2 - [ (3x)2 - 2.3.x.2y + ( 2x)2
= ( x + y +z )2 - ( 3y - 2x)2
= ( x + y +z - 3y + 2x )(x+ y + z + 3y - 2x )