1: 3x(x-2y)+6y(2y-x)
=3x(x-2y)-6y(x-2y)
=(x-2y)(3x-6y)
=3(x-2y)(x-2y)
\(=3\left(x-2y\right)^2\)
2: \(a\cdot x^2-ax+bx^2-bx+a+b\)
\(=x^2\left(a+b\right)-x\left(a+b\right)+\left(a+b\right)\)
\(=\left(a+b\right)\left(x^2-x+1\right)\)
3: \(81x^2-6yz-9y^2-z^2\)
\(=\left(9x\right)^2-\left(9y^2+6yz+z^2\right)\)
\(=\left(9x\right)^2-\left(3y+z\right)^2\)
=(9x-3y-z)(9x+3y+z)
4: \(x^4+1-2x^2=x^4-2x^2+1=\left(x^2-1\right)^2\)
\(=\left(x-1\right)^2\cdot\left(x+1\right)^2\)
5: \(2x^3-x^2-8x+4\)
\(=x^2\left(2x-1\right)-4\left(2x-1\right)\)
\(=\left(2x-1\right)\left(x^2-4\right)=\left(2x-1\right)\left(x-2\right)\left(x+2\right)\)
6: \(x\left(x-y\right)^2+y\left(x-y\right)^2-xy+x^2\)
\(=\left(x-y\right)^2\cdot\left(x+y\right)+x\left(x-y\right)\)
=(x-y)[(x-y)(x+y)+x]
\(=\left(x-y\right)\left(x^2-y^2+x\right)\)
7: \(x\left(x-y\right)^2+y\left(x-y\right)^2-xy+x^2\)
\(=\left(x-y\right)^2\cdot\left(x+y\right)+x\left(x-y\right)\)
=(x-y)[(x-y)(x+y)+x]
\(=\left(x-y\right)\left(x^2-y^2+x\right)\)
8: \(x^3-4x^2-8x+32\)
\(=x^2\left(x-4\right)-8\left(x-4\right)\)
\(=\left(x-4\right)\left(x^2-8\right)\)
9: \(x^2-2xy+y^2-z^2+2zt-t^2\)
\(=\left(x-y\right)^2-\left(z-t\right)^2\)
=(x-y+z-t)(x-y-z+t)
10: \(\left(a+b+c\right)^3-a^3-b^3-c^3\)
\(=\left(a+b+c-a\right)\left\lbrack\left(a+b+c\right)^2+a\left(a+b+c\right)+a^2\right\rbrack-\left(b+c\right)\left(b^2-bc+c^2\right)\)
\(=\left(b+c\right)\left(a^2+b^2+c^2+2ab+2ac+2bc+a^2+ab+ac+a^2\right)-\left(b+c\right)\left(b^2-bc+c^2\right)\)
=(b+c)\(\left(3a^2+b^2+c^2+3ab+3ac+2bc-b^2+bc-c^2\right)\)
=(b+c)\(\left(3a^2+3ab+3ac+3bc\right)\)
=3(b+c)[a(a+b)+c(a+b)]
=3(b+c)(a+b)(a+c)
11: \(x^2-y^2-5y+5x\)
=(x-y)(x+y)+5(x-y)
=(x-y)(x+y+5)
12: \(4x^2-16x^2y^2+y^2+4xy\)
\(=\left(4x^2+4xy+y^2\right)-\left(4xy\right)^2\)
\(=\left(2x+y\right)^2-\left(4xy\right)^2\)
=(2x+y-4xy)(2x+y+4xy)
13: \(x^2-8x+7\)
\(=x^2-x-7x+7\)
=x(x-1)-7(x-1)
=(x-1)(x-7)
14: \(2x^2-5x+2\)
\(=2x^2-4x-x+2\)
=2x(x-2)-(x-2)
=(x-2)(2x-1)
15: (x+1)(x+2)(x+3)(x+4)-8
\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-8\)
\(=\left(x^2+5x+5\right)^2-1-8\)
\(=\left(x^2+5x+5\right)^2-9\)
\(=\left(x^2+5x+5+3\right)\left(x^2+5x+5-3\right)\)
\(=\left(x^2+5x+8\right)\left(x^2+5x+2\right)\)
16: \(x^4+1024\)
\(=x^4+64x^2+1024-64x^2\)
\(=\left(x^2+32\right)^2-\left(8x\right)^2=\left(x^2-8x+32\right)\left(x^2+8x+32\right)\)
17: \(x^3+9x^2-4x-36\)
\(=x^2\left(x+9\right)-4\left(x+9\right)\)
\(=\left(x+9\right)\left(x^2-4\right)=\left(x+9\right)\left(x-2\right)\left(x+2\right)\)
18: \(y^2-4x^2+4x-1\)
\(=y^2-\left(4x^2-4x+1\right)\)
\(=y^2-\left(2x-1\right)^2\)
=(y-2x+1)(y+2x-1)
19: \(x^3-16x-15x\left(x-4\right)\)
\(=x\left(x^2-16\right)-15x\left(x-4\right)\)
=x(x-4)(x+4)-15x(x-4)
=x(x-4)(x+4-15)
=x(x-4)(x-11)
20: \(x^3-5x^2+8x-4\)
\(=x^3-x^2-4x^2+4x+4x-4\)
\(=x^2\left(x-1\right)-4x\left(x-1\right)+4\left(x-1\right)\)
\(=\left(x-1\right)\left(x^2-4x+4\right)=\left(x-1\right)\left(x-2\right)^2\)
21: \(x^4+64\)
\(=x^4+16x^2+64-16x^2\)
\(=\left(x^2+8\right)^2-\left(4x\right)^2\)
\(=\left(x^2+8+4x\right)\left(x^2+8-4x\right)\)
