1: \(x^2-y^2-x+y\)
\(=\left(x^2-y^2\right)-\left(x-y\right)\)
\(=\left(x-y\right)\left(x+y\right)-\left(x-y\right)\)
\(=\left(x-y\right)\left(x+y-1\right)\)
2: \(x^2-4-2\left(x-2\right)^2\)
\(=\left(x^2-4\right)-\left(x-2\right)\left(2x-4\right)\)
\(=\left(x-2\right)\left(x+2\right)-\left(x-2\right)\left(2x-4\right)\)
\(=\left(x-2\right)\left(x+2-2x+4\right)\)
\(=\left(-x+6\right)\left(x-2\right)\)
3: \(x^3-x+y^3-y\)
\(=\left(x^3+y^3\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2-1\right)\)
4: \(x^2-y^2-2x-2y\)
\(=\left(x^2-y^2\right)-\left(2x+2y\right)\)
\(=\left(x-y\right)\left(x+y\right)-2\left(x+y\right)\)
\(=\left(x+y\right)\left(x-y-2\right)\)
5: \(x^2-9-x^2\left(x^2-9\right)\)
\(=\left(x^2-9\right)-x^2\left(x^2-9\right)\)
\(=\left(x^2-9\right)\left(1-x^2\right)\)
\(=\left(x-3\right)\left(x+3\right)\left(1-x\right)\left(1+x\right)\)
6: \(4x^2-9-\left(3x+1\right)\left(2x-3\right)\)
\(=\left(4x^2-9\right)-\left(3x+1\right)\left(2x-3\right)\)
\(=\left(2x+3\right)\left(2x-3\right)-\left(3x+1\right)\left(2x-3\right)\)
\(=\left(2x-3\right)\left(2x+3-3x-1\right)\)
\(=\left(-x+2\right)\left(2x-3\right)\)
