a: \(3x^4+9x^3y-15x^2y^2-45xy^3\)
\(=3x\left(x^3+3x^2y-5xy^2-15y^2\right)\)
\(=3x\left[x^2\left(x+3y\right)-5y^2\left(x+3y\right)\right]\)
\(=3x\left(x+3y\right)\left(x^2-5y^2\right)\)
b: \(x^3-8-3\left(x-1\right)\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2+2x+4\right)-\left(x-2\right)\left(3x-3\right)\)
\(=\left(x-2\right)\left(x^2+2x+4-3x+3\right)=\left(x-2\right)\left(x^2-x+7\right)\)
c: \(9x^2\left(2x-3\right)-12xy\left(3-2x\right)-4y^2\left(3-2x\right)\)
\(=9x^2\left(2x-3\right)+12xy\left(2x-3\right)+4y^2\left(2x-3\right)\)
\(=\left(2x-3\right)\left(9x^2+12xy+4y^2\right)=\left(2x-3\right)\left(3x+2y\right)^2\)
d: \(\left(3x-11\right)\left(4x^2-1\right)-\left(9x-3\right)\left(4x^2-4x+1\right)\)
\(=\left(3x-11\right)\left(2x-1\right)\left(2x+1\right)-\left(9x-3\right)\left(2x-1\right)^2\)
\(=\left(2x-1\right)\left[\left(3x-11\right)\left(2x+1\right)-\left(9x-3\right)\left(2x-1\right)\right]\)
\(=\left(2x-1\right)\left(6x^2-19x-11-18x^2+15x-3\right)\)
\(=\left(2x-1\right)\left(-12x^2-4x-14\right)\)
\(=-2\left(2x-1\right)\left(6x^2+2x+7\right)\)
