1: \(x^3-2x^2=x^2\left(x-2\right)\)
2: \(x^3-16x=x\left(x^2-16\right)=x\left(x-4\right)\left(x+4\right)\)
3: \(12x^3y^3z^2-27x^3y^4z^3=3x^3y^3z^2\left(4-9yz\right)\)
4:x(x+1)+(x+1)(x+2)
=(x+1)(x+x+2)
=(x+1)(2x+2)
=2(x+1)(x+1)
\(=2\left(x+1\right)^2\)
5: 4(x-2)(x+3)-2x(x+3)
=(x+3)(4x-8-2x)
=(x+3)(2x-8)
=2(x-4)(x+3)
6: \(x^2-y^2+\left(x-y\right)\left(2x+y\right)\)
=(x-y)(x+y)+(x-y)(2x+y)
=(x-y)(x+y+2x+y)
=(x-y)(3x+2y)
7: \(x^2\left(y+3\right)-\left(y+3\right)=\left(y+3\right)\left(x^2-1\right)\)
=(y+3)(x-1)(x+1)
8: 5x(2x+3y)-2x-3y
=5x(2x+3y)-(2x+3y)
=(2x+3y)(5x-1)
9: (4xy-1)(2x+3)-4xy+1
=(4xy-1)(2x+3)-(4xy-1)
=(4xy-1)(2x+3-1)
=(4xy-1)(2x+2)
=2(x+1)(4xy-1)
10: (2x+1)(x-3)-3x(3-x)
=(2x+1)(x-3)+3x(x-3)
=(x-3)(2x+1+3x)
=(5x+1)(x-3)
12: (4-3x)(x+5)+(3x-4)(2x-1)
=(3x-4)(-x-5+2x-1)
=(3x-4)(x-6)
13: 5x-2y-(2x+1)(2y-5x)
=(5x-2y)+(2x+1)(5x-2y)
=(5x-2y)(2x+1+1)
=(2x+2)(5x-2y)
=2(x+1)(5x-2y)
14: 3-2x+(2x-1)(2x-3)
=-(2x-3)+(2x-1)(2x-3)
=(2x-3)(-1+2x-1)
=(2x-3)(2x-2)
=2(x-1)(2x-3)
15: \(\left(2-x\right)^2+\left(x+1\right)\left(x-2\right)\)
\(=\left(x-2\right)^2+\left(x+1\right)\left(x-2\right)\)
=(x-2)(x-2+x+1)
=(2x-1)(x-2)
16: \(\left(3x+2\right)^2-\left(3x+2\right)\left(x-1\right)\)
=(3x+2)(3x+2-x+1)
=(2x+3)(3x+2)
17: \(\left(x-5\right)^2+\left(3x-1\right)\left(5-x\right)\)
\(=\left(5-x\right)^2+\left(3x-1\right)\left(5-x\right)\)
=(5-x)(5-x+3x-1)
=(5-x)(2x+4)
=2(x+2)(5-x)
18: \(4x^2-4x-y^2+1\)
\(=\left(4x^2-4x+1\right)-y^2\)
\(=\left(2x-1\right)^2-y^2\)
=(2x-1-y)(2x-1+y)
19: \(x^2-2xy+y^2-z^2+2zt-t^2\)
\(=\left(x^2-2xy+y^2\right)-\left(z^2-2zt+t^2\right)\)
\(=\left(x-y\right)^2-\left(z-t\right)^2\)
=(x-y-z+t)(x-y+z-t)
20: \(6x^2+7x-3\)
\(=6x^2+9x-2x-3\)
=3x(2x+3)-(2x+3)
=(2x+3)(3x-1)
11: (5-3x)(2x+3)-15+9x
=(-2x-3)(3x-5)+3(3x-5)
=(3x-5)(-2x-3+3)
=-2x(3x-5)
