a) \(x^3-3x^2+1-3x\)
\(=\left(x^3+1\right)-\left(3x^2+3x\right)\)
\(=\left(x+1\right)\left(x^2-x+1\right)-3x\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-x+1-3x\right)\)
\(=\left(x+1\right)\left(x^2-4x+1\right)\)
\(=\left(x+1\right)\left(x^2-4x+4-5\right)\)
\(=\left(x+1\right)\left[\left(x-2\right)^2-5\right]\)
\(=\left(x+1\right)\left(x-2-\sqrt{5}\right)\left(x-2+\sqrt{5}\right)\)
b) \(x^2-6xy-25z^2+9y^2\)
\(=\left(x^2-6xy+9y^2\right)-25z^2\)
\(=\left(x-3y\right)^2-25z^2\)
\(=\left(x-3y-5z\right)\left(x-3y+5z\right)\)
c) \(x^2-y^2-x-y\)
\(=\left(x-y\right)\left(x+y\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x-y-1\right)\)
d) \(x^2-y^2+4-4x\)
\(=\left(x^2-4x+4\right)-y^2\)
\(=\left(x-2\right)^2-y^2\)
\(=\left(x-2-y\right)\left(x-2+y\right)\)
e) \(a^3-ay-a^2x+xy\)
\(=a\left(a^2-y\right)-x\left(a^2-y\right)\)
\(=\left(a^2-y\right)\left(a-x\right)\)
