a: \(3x-3y+x^2-y^2\)
\(=\left(3x-3y\right)+\left(x^2-y^2\right)\)
\(=3\left(x-y\right)+\left(x-y\right)\left(x+y\right)\)
\(=\left(x-y\right)\left(x+y+3\right)\)
b: \(x^2-4x^2y^2+y^2+2xy\)
\(=\left(x^2+2xy+y^2\right)-4x^2y^2\)
\(=\left(x+y\right)^2-\left(2xy\right)^2\)
\(=\left(x+y+2xy\right)\left(x+y-2xy\right)\)
c: \(x^6-x^4+2x^3+2x^2\)
\(=x^2\left(x^4-x^2+2x+2\right)\)
\(=x^2\left[x^2\left(x^2-1\right)+2\left(x+1\right)\right]\)
\(=x^2\left[x^2\left(x-1\right)\left(x+1\right)+2\left(x+1\right)\right]\)
\(=x^2\left(x+1\right)\left(x^3-x^2+2\right)\)
d: \(x^3-3x^2+3x-1-y^3\)
\(=\left(x^3-3x^2+3x-1\right)-y^3\)
\(=\left(x-1\right)^3-y^3\)
\(=\left(x-1-y\right)\left[\left(x-1\right)^2+y\left(x-1\right)+y^2\right]\)
\(=\left(x-y-1\right)\left(x^2-2x+1+xy-y+y^2\right)\)
e: \(x^2\left(x-3\right)^2-\left(x-3\right)^2-x^2+1\)
\(=\left[x^2\left(x-3\right)^2-\left(x-3\right)^2\right]-\left(x^2-1\right)\)
\(=\left(x-3\right)^2\left(x^2-1\right)-\left(x^2-1\right)\)
\(=\left(x^2-1\right)\left[\left(x-3\right)^2-1\right]\)
\(=\left(x-1\right)\left(x+1\right)\left(x-3-1\right)\left(x-3+1\right)\)
\(=\left(x-1\right)\left(x+1\right)\left(x-4\right)\left(x-2\right)\)
f: \(x^3-2x^2+4x-8\)
\(=\left(x^3-2x^2\right)+\left(4x-8\right)\)
\(=x^2\left(x-2\right)+4\left(x-2\right)=\left(x-2\right)\left(x^2+4\right)\)
g: \(\left(x+y\right)^3-\left(x-y\right)^3\)
\(=\left[\left(x+y\right)-\left(x-y\right)\right]\left[\left(x+y\right)^2+\left(x+y\right)\left(x-y\right)+\left(x-y\right)^2\right]\)
\(=\left(x+y-x+y\right)\left(x^2+2xy+y^2+x^2-y^2+x^2-2xy+y^2\right)\)
\(=2y\cdot\left(3x^2+y^2\right)\)
h: \(2a^2\left(x+y+z\right)-4ab\left(x+y+z\right)+2b^2\left(x+y+z\right)\)
\(=2\left(x+y+z\right)\left(a^2-2ab+b^2\right)\)
\(=2\left(x+y+z\right)\left(a-b\right)^2\)
