16: \(x^2+2x+1-y^2\)
\(=\left(x^2+2x+1\right)-y^2\)
\(=\left(x+1\right)^2-y^2\)
\(=\left(x+1+y\right)\left(x+1-y\right)\)
17: \(x^2-2x+1-4y^2\)
\(=\left(x^2-2x+1\right)-4y^2\)
\(=\left(x-1\right)^2-\left(2y\right)^2\)
\(=\left(x-1-2y\right)\left(x-1+2y\right)\)
18: \(x^2-1+4y-4y^2\)
\(=x^2-\left(4y^2-4y+1\right)\)
\(=x^2-\left(2y-1\right)^2\)
\(=\left(x-2y+1\right)\left(x+2y-1\right)\)
19: \(4x^2+4x-9y^2+1\)
\(=\left(4x^2+4x+1\right)-9y^2\)
\(=\left(2x+1\right)^2-\left(3y\right)^2\)
\(=\left(2x+1+3y\right)\left(2x+1-3y\right)\)
20: \(x^2+4y^2+4xy-16\)
\(=\left(x^2+4xy+4y^2\right)-16\)
\(=\left(x+2y\right)^2-4^2\)
\(=\left(x+2y-4\right)\left(x+2y+4\right)\)
21: \(9-x^2+6xy-9y^2\)
\(=9-\left(x^2-6xy+9y^2\right)\)
\(=3^2-\left(x-3y\right)^2\)
\(=\left(3-x+3y\right)\left(3+x-3y\right)\)
22: \(x^2-4y^2+4y-1\)
\(=x^2-\left(4y^2-4y+1\right)\)
\(=x^2-\left(2y-1\right)^2\)
\(=\left[x-\left(2y-1\right)\right]\left(x+2y-1\right)\)
\(=\left(x-2y+1\right)\left(x+2y-1\right)\)
23: \(25x^2+4ab-4b^2-a^2\)
\(=25x^2-\left(a^2-4ab+4b^2\right)\)
\(=\left(5x\right)^2-\left(a-2b\right)^2\)
\(=\left[5x-\left(a-2b\right)\right]\left[5x+\left(a-2b\right)\right]\)
\(=\left(5x-a+2b\right)\left(5x+a-2b\right)\)
24: \(25x^2+1-9y^2-10x\)
\(=\left(25x^2-10x+1\right)-9y^2\)
\(=\left(5x-1\right)^2-\left(3y\right)^2\)
\(=\left(5x-1-3y\right)\left(5x-1+3y\right)\)
25: \(3x^2+6ax+3a^2-3b^2\)
\(=3\left(x^2+2ax+a^2-b^2\right)\)
\(=3\left[\left(x^2+2ax+a^2\right)-b^2\right]\)
\(=3\left[\left(x+a\right)^2-b^2\right]\)
\(=3\left(x+a-b\right)\left(x+a+b\right)\)
26: \(3a^2+6ab+3b^2-12c^2\)
\(=3\left[a^2+2ab+b^2-4c^2\right]\)
\(=3\left[\left(a^2+2ab+b^2\right)-\left(2c\right)^2\right]\)
\(=3\left[\left(a+b\right)^2-\left(2c\right)^2\right]\)
\(=3\left(a+b-2c\right)\left(a+b+2c\right)\)
27: \(x^4+x^3+x+1\)
\(=\left(x^4+x^3\right)+\left(x+1\right)\)
\(=x^3\left(x+1\right)+\left(x+1\right)\)
\(=\left(x+1\right)\left(x^3+1\right)\)
\(=\left(x+1\right)\left(x+1\right)\left(x^2-x+1\right)\)
\(=\left(x+1\right)^2\cdot\left(x^2-x+1\right)\)
28: \(x^4-9x^3+x^2-9x\)
\(=\left(x^4-9x^3\right)+\left(x^2-9x\right)\)
\(=x^3\left(x-9\right)+x\left(x-9\right)\)
\(=\left(x-9\right)\left(x^3+x\right)\)
\(=x\left(x^2+1\right)\left(x-9\right)\)
29: \(x^4-9x^3+x^2-9x\)
\(=\left(x^4-9x^3\right)+\left(x^2-9x\right)\)
\(=x^3\left(x-9\right)+x\left(x-9\right)\)
\(=\left(x-9\right)\left(x^3+x\right)=x\left(x^2+1\right)\left(x-9\right)\)
15: \(-2y-4y^2+4yz-z^2+z\)
\(=-\left(2y+4y^2-4yz+z^2-z\right)\)
\(=-\left[\left(2y-z\right)+\left(4y^2-4yz+z^2\right)\right]\)
\(=-\left[\left(2y-z\right)+\left(2y-z\right)^2\right]\)
\(=-\left(2y-z\right)\left(1+2y-z\right)\)
14: \(7y-4xy+14x-y^2-4x^2\)
\(=\left(14x+7y\right)+\left(4x^2+4xy+y^2\right)\)
\(=7\left(2x+y\right)+\left(2x+y\right)^2\)
\(=\left(2x+y\right)\left(2x+y+7\right)\)
13: \(9x^2+15x+5y+6xy+y^2\)
\(=\left(9x^2+6xy+y^2\right)+\left(15x+5y\right)\)
\(=\left(3x+y\right)^2+5\left(3x+y\right)\)
\(=\left(3x+y\right)\left(3x+y+5\right)\)
12: \(x^2-4xy+4y^2+3x-6y\)
\(=\left(x^2-4xy+4y^2\right)+\left(3x-6y\right)\)
\(=\left(x-2y\right)^2+3\left(x-2y\right)\)
\(=\left(x-2y\right)\left(x-2y+3\right)\)
11: \(3a-3b-a^2+2ab-b^2\)
\(=\left(3a-3b\right)-\left(a^2-2ab+b^2\right)\)
\(=3\left(a-b\right)-\left(a-b\right)^2\)
\(=\left(a-b\right)\left(3-a+b\right)\)
10: \(x^2+4x-2xy-4y+y^2\)
\(=\left(x^2-2xy+y^2\right)+\left(4x-4y\right)\)
\(=\left(x-y\right)^2+4\left(x-y\right)\)
\(=\left(x-y\right)\left(x-y+4\right)\)
9: \(27x^3-3x+2y-8y^3\)
\(=\left(27x^3-8y^3\right)-\left(3x-2y\right)\)
\(=\left(3x-2y\right)\left[9x^2+6xy+4y^2\right]-\left(3x-2y\right)\)
\(=\left(3x-2y\right)\left(9x^2+6xy+4y^2-1\right)\)
8: \(x^3-2x+y^3-2y\)
\(=\left(x^3+y^3\right)-\left(2x+2y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)-2\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2-2\right)\)
7: \(8x^3-y^3-4x+2y\)
\(=\left(8x^3-y^3\right)-\left(4x-2y\right)\)
\(=\left(2x-y\right)\left(4x^2+2xy+y^2\right)-2\left(2x-y\right)\)
\(=\left(2x-y\right)\left(4x^2+2xy+y^2-2\right)\)
6: \(4x^2-9-\left(3x+1\right)\left(2x-3\right)\)
\(=\left(2x-3\right)\left(2x+3\right)-\left(3x+1\right)\left(2x-3\right)\)
\(=\left(2x-3\right)\left(2x+3-3x-1\right)\)
\(=\left(2x-3\right)\left(-x+2\right)\)
