a: \(x^2-3x+xy-3y\)
\(=\left(x^2-3x\right)+\left(xy-3y\right)\)
\(=x\left(x-3\right)+y\left(x-3\right)\)
\(=\left(x-3\right)\left(x+y\right)\)
b: \(3x^2-3xy-5x+5y\)
\(=\left(3x^2-3xy\right)-\left(5x-5y\right)\)
\(=3x\left(x-y\right)-5\left(x-y\right)\)
\(=\left(x-y\right)\left(3x-5\right)\)
c: \(x^2-2xy+y^2-z^2+2zt-t^2\)
\(=\left(x^2-2xy+y^2\right)-\left(z^2-2zt+t^2\right)\)
\(=\left(x-y\right)^2-\left(z-t\right)^2\)
\(=\left(x-y-z+t\right)\left(x-y+z-t\right)\)
d: \(3x^2+6xy+3y^2-3z^2\)
\(=3\left(x^2+2xy+y^2-z^2\right)\)
\(=3\left[\left(x^2+2xy+y^2\right)-z^2\right]\)
\(=3\left[\left(x+y\right)^2-z^2\right]\)
\(=3\left(x+y+z\right)\left(x+y-z\right)\)
e: \(27x^3-54x^2y+36xy^2-8y^3\)
\(=\left(3x\right)^3-3\left(3x\right)^2\cdot2y+3\cdot3x\cdot\left(2y\right)^2-\left(2y\right)^3\)
\(=\left(3x-2y\right)^3\)
f: \(a^3-1+5a^2-5+3a-3\)
\(=\left(a^3-1\right)+\left(5a^2-5\right)+\left(3a-3\right)\)
\(=\left(a-1\right)\left(a^2+a+1\right)+5\left(a-1\right)\left(a+1\right)+3\left(a-1\right)\)
\(=\left(a-1\right)\left(a^2+a+1+5a+5+3\right)\)
\(=\left(a-1\right)\left(a^2+6a+9\right)=\left(a-1\right)\cdot\left(a+3\right)^2\)
