a: \(2x+10=2\cdot x+2\cdot5=2\left(x+5\right)\)
\(x^2+xy+x=x\left(x+y+1\right)\)
\(3x^2y-6xy+12xy^2=3xy\cdot x-3xy\cdot2+3xy\cdot4y=3xy\left(x-2+4y\right)\)
b: y(x-2)-2x(x-2)
=(x-2)(y-2x)
\(2x^2\left(x-2y\right)+xy\left(x-2y\right)\)
\(=\left(x-2y\right)\left(2x^2+xy\right)\)
=x(2x+y)(x-2y)
\(x\left(y-x\right)-2y\left(x-y\right)\)
=x(y-x)+2y(y-x)
=(y-x)(x+2y)
c: \(x^2-9=\left(x-3\right)\left(x+3\right)\)
\(\left(4x-1\right)^2-4x^2\)
=(4x-1-2x)(4x-1+2x)
=(2x-1)(6x-1)
\(\left(x+1\right)^2-\left(2x-1\right)^2\)
=(x+1+2x-1)(x+1-2x+1)
\(=3x\left(-x+2\right)\)
\(x^2+4x+4=x^2+2\cdot x\cdot2+2^2=\left(x+2\right)^2\)
d: \(9x^2-6x+1=\left(3x\right)^2-2\cdot3x\cdot1+1^2=\left(3x-1\right)^2\)
\(4x-4-x^2=-\left(x^2-4x+4\right)=-\left(x-2\right)^2\)
\(8x^3-1=\left(2x-1\right)\left(4x^2+2x+1\right)\)
\(8x^3-12x^2y+6xy^2-y^3\)
\(=\left(2x\right)^3-3\cdot\left(2x\right)^2\cdot y+3\cdot2x\cdot y^2-y^3\)
\(=\left(2x-y\right)^3\)
e: \(x^4-2x^3+x^2-2x\)
\(=x^3\left(x-2\right)+x\left(x-2\right)\)
\(=\left(x-2\right)\left(x^3+x\right)=x\left(x^2+1\right)\left(x-2\right)\)
\(xy+1-x^2+y\)
=y(x+1)+(1-x)(1+x)
=(x+1)(y+1-x)
f: \(x^2-2x+1-y^2\)
\(=\left(x-1\right)^2-y^2\)
=(x-1-y)(x-1+y)
\(x^2-2xy+y^2-9z^2\)
\(=\left(x-y\right)^2-\left(3z\right)^2\)
=(x-y-3z)(x-y+3z)
\(x^2-4xy-z^2+4y^2\)
\(=\left(x^2-4xy+4y^2\right)-z^2\)
\(=\left(x-2y\right)^2-z^2\)
=(x-2y-z)(x-2y+z)
g: x(x-1)+x(x+3)
=x(x-1+x+3)
=x(2x+2)
=2x(x+1)
\(3xy^2-6xy+3x\)
\(=3x\left(y^2-2y+1\right)=3x\left(y-1\right)^2\)
xy+2x+y+2
=x(y+2)+(y+2)
=(x+1)(y+2)
