1: \(x^3+2x^2+x\)
\(=x\cdot x^2+x\cdot2x+x\cdot1\)
\(=x\left(x^2+2x+1\right)=x\left(x+1\right)^2\)
2: \(x^4-4x^3+4x^2\)
\(=x^2\cdot x^2-x^2\cdot4x+x^2\cdot4\)
\(=x^2\left(x^2-4x+4\right)\)
\(=x^2\left(x-2\right)^2\)
3: \(5x^3-10x^2+5x\)
\(=5x\cdot x^2-5x\cdot2x+5x\cdot1\)
\(=5x\left(x^2-2x+1\right)\)
\(=5x\left(x-1\right)^2\)
4: \(2x^3-12x^2+18x\)
\(=2x\cdot x^2-2x\cdot6x+2x\cdot9\)
\(=2x\left(x^2-6x+9\right)=2x\left(x-3\right)^2\)
5: \(8x^2y-8xy+2x\)
\(=2x\cdot4xy-2x\cdot4y+2x\cdot1\)
\(=2x\left(4xy-4y+1\right)\)
6: \(5x^2y-35xy+60y\)
\(=5y\cdot x^2-5y\cdot7x+5y\cdot12\)
\(=5y\left(x^2-7x+12\right)\)
\(=5y\left(x-4\right)\left(x-3\right)\)
7: \(2x^2+5x^3+x^2y\)
\(=x^2\cdot5x+x^2\cdot2+x^2\cdot y\)
\(=x^2\left(5x+2+y\right)\)
8: \(2x^3y-8x^2y+8xy\)
\(=2xy\cdot x^2-2xy\cdot4x+2xy\cdot4\)
\(=2xy\left(x^2-4x+4\right)=2xy\left(x-2\right)^2\)
9: \(4x^2y-8xy^2+18x^2y^2\)
\(=2xy\cdot2x-2xy\cdot4y+2xy\cdot9xy\)
\(=2xy\cdot\left(2x-4y+9xy\right)\)
