1)
x^3 - x^2 - 4x^2 + 8x - 4=x^3-4x^2+4x-x^2+4x-4
=x(x^2-4x+4)-(x^2-4x+4)
=(x-1)(x^2-4x+4)
2)
4x^2 - 25 - (2x - 5)(2x + 7)
=(2x-5)(2x+5)-(2x-5)(2x+7)
=(2x-5)(2x+5-2x-7)
=-2(2x-5)
3)
x^3 + 27 + (x + 3)(x - 9)
=(x+3)(x^2-3x+9)+(x+3)(x-9)
=(x+3)(x^2-3x+9+x-9)
=(x+3)(x^2-2x)=x(x+3)(x-2)
4)
4x^2y^2 - (x^2 + y^2 - z^2)^2
=(2xy-x^2-y^2+z^2)(2xy+x^2+y^2-z^2)
=[ (x-y)^2-z^2] [ (x+y)^2-z^2]
=(x-y-z)(x-y+z)(x+y-z)(x+y+z)
a) \(x^3-x^2-4x^2+8x-4\)
\(=x^3-4x^2+4x-x^2+4x-4\)
\(=x\left(x^2-4x+4\right)-\left(x^2-4x+4\right)\)
\(=\left(x-1\right)\left(x^2-4x+4\right)\)
\(=\left(x-1\right)\left(x-2\right)^2\)
b) \(4x^2-25-\left(2x-5\right)\left(2x+7\right)\)
\(=\left(2x-5\right)\left(2x+5\right)-\left(2x-5\right)\left(2x+7\right)\)
\(=\left(2x-5\right)\left(2x+5-2x-7\right)\)
\(=2\left(2x-5\right)\)
c) \(x^3+27+\left(x+3\right)\left(x-9\right)\)
\(=\left(x+3\right)\left(x^2-3x+9\right)+\left(x+3\right)\left(x-9\right)\)
\(=\left(x+3\right)\left(x^2-3x+9+x-9\right)\)
\(=\left(x+3\right)\left(x^2-2x\right)\)
\(=x\left(x+3\right)\left(x-2\right)\)
d) \(4x^2y^2-\left(x^2+y^2-z^2\right)^2\)
\(=\left(2xy-x^2-y^2+z^2\right)\left(2xy+x^2+y^2-z^2\right)\)
\(=\left[\left(x-y\right)^2-z^2\right]\left[\left(x+y\right)^2-z^2\right]\)
\(=\left(x-y-z\right)\left(x-y+z\right)\left(x+y-z\right)\left(x+y+z\right).\)