\(\left(xy+1\right)^2-\left(x+y\right)^2=\left(xy+1-x-y\right)\left(xy+1+x+y\right)=\left[x\left(y-1\right)-\left(y-1\right)\right]\left[x\left(y+1\right)+\left(y+1\right)\right]=\left(x-1\right)\left(y-1\right)\left(x+1\right)\left(y+1\right)\)

\(\left(xy+1\right)^2-\left(x+y\right)^2\)

\(=\left(xy-x-y+1\right)\left(xy+1+x+y\right)\)

\(=\left(y-1\right)\left(x-1\right)\left(y+1\right)\left(x+1\right)\)

x²y + xy² - x - y

= (x²y + xy²) - (x + y)

= xy(x + y) - (x + y)

= (x + y)(xy - 1)

\(x^2-xy\left(a+b\right)+aby^2=x^2-xya-xyb+aby^2=x\left(x-ya\right)-yb\left(x-ya\right)=\left(x-ya\right)\left(x-yb\right)\)

\(x^2-xy\left(a+b\right)+aby^2\)

\(=x^2-axy-bxy+aby^2\)

\(=x\left(x-ay\right)-by\left(x-ay\right)\)

\(=\left(x-ay\right)\left(x-by\right)\)

2: \(8xy-24xy+16x\)

\(=8x\cdot y-8x\cdot3y+8x\cdot2\)

\(=8x\left(y-3y+2\right)=8x\left(-2y+2\right)\)

\(=-16y\left(y-1\right)\)

3: \(xy-x=x\cdot y-x\cdot1=x\left(y-1\right)\)

11: \(2mx-4m2xy+6mx\)

\(=2mx-2my\cdot4y+2mx\cdot3\)

\(=2mx\left(1-4y+3\right)\)

\(=2mx\left(4-4y\right)=8mx\left(1-y\right)\)

12: \(7x^2y^5-14x^3y^4-21y^3\)

\(=7y^3\cdot x^2y^2-7y^3\cdot2x^3y-7y^3\cdot3\)

\(=7y^3\left(x^2y^2-2x^3y-3\right)\)

13: \(2\left(x-y\right)-a\left(x-y\right)\)

\(=2\cdot\left(x-y\right)-a\cdot\left(x-y\right)\)

\(=\left(x-y\right)\left(2-a\right)\)

\(\left(x+y\right)^2+3\left(x+y\right)-10=\left[\left(x+y\right)^2+2\left(x+y\right).\dfrac{3}{2}+\dfrac{9}{4}\right]-\dfrac{49}{4}\)

\(=\left(x+y+\dfrac{3}{2}\right)^2-\dfrac{49}{4}=\left(x+y+\dfrac{3}{2}-\dfrac{7}{2}\right)\left(x+y+\dfrac{3}{2}+\dfrac{7}{2}\right)=\left(x+y-2\right)\left(x+y+5\right)\)

\(\left(x+y\right)^2+3\left(x+y\right)-10\)

\(=\left(x+y\right)^2+5\left(x+y\right)-2\left(x+y\right)-10\)

\(=\left(x+y+5\right)\left(x+y-2\right)\)

\(=x\left[x^2\left(x-y\right)^2-36y^2\right]\\ =x\left[x\left(x-y\right)-6y\right]\left[x\left(x-y\right)+6y\right]\\ =x\left(x^2-xy-6y\right)\left(x^2-xy+6y\right)\)

\(\left(x+y+z\right)^2+\left(x+y-z\right)^2-4z^2=\left(x+y+z\right)^2+\left(x+y-z-2z\right)\left(x+y-z+2z\right)=\left(x+y+z\right)^2+\left(x+y-3z\right)\left(x+y+z\right)=\left(x+y+z\right)\left(x+y+z+x+y-3z\right)=\left(x+y+z\right)\left(2x+2y-2z\right)=2\left(x+y+z\right)\left(x+y-z\right)\)

Ta có:

 (x + y + z)² + (x + y – z)² – 4z²

^{\(=\left(x+y-z\right)^2+\left(x+y-z\right)\left(x+y+3z\right)\)}

​\(=\left(x+y-z\right)\left(x+y+3z+x+y-z\right)\)​

\(\left(x+y+z\right)^2+\left(x+y-z\right)^2-4z^2\)

\(=x^2+y^2+z^2+2xy+2yz+2xz+x^2+y^2+z^2+2xy-2xz-2yz-4z^2\)

\(=2x^2+2y^2-2z^2+4xy\)

\(=2\left(x^2+2xy+y^2-z^2\right)\)

\(=2\left(x+y-z\right)\left(x+y+z\right)\)

ab(x²+y²)+xy(a²+b²)

\(=abx^2+aby^2+a^2xy+b^2xy=\left(abx^2+a^2xy\right)+\left(aby^2+b^2xy\right).\)

\(=ax\left(bx+ay\right)+by\left(ay+bx\right)=\left(ax+by\right).\left(ay+bx\right)\)