a: \(3x-15y=3\cdot x-3\cdot5y=3\left(x-5y\right)\)
\(21x^2y-12xy^2=3xy\cdot7x-3xy\cdot4y=3xy\left(7x-4y\right)\)
\(x^3+x^2-2x\)
\(=x\left(x^2+x-2\right)=x\left(x^2+2x-x-2\right)\)
=x(x+2)(x-1)
b: \(3x\left(x-1\right)+7x^2\left(x-1\right)=\left(x-1\right)\left(7x^2+3x\right)=x\left(7x+3\right)\left(x-1\right)\)
\(3x\left(x-a\right)+4a\left(x-a\right)=\left(x-a\right)\left(3x+4a\right)\)
c: \(\frac12x\left(x^2-4\right)+4\left(x+2\right)\)
\(=\left(x+2\right)\left\lbrack\frac12x\left(x-2\right)+4\right\rbrack=\left(x+2\right)\left(\frac12x^2-x+4\right)\)
d: \(6x^4-9x^3=3x^3\cdot2x-3x^3\cdot3=3x^3\left(2x-3\right)\)
\(5y^{10}+15y^6=5y^6\left(y^4+3\right)\)
e: \(9x^2y^2+15x^2y-21xy^2\)
\(=3xy\cdot3xy+3xy\cdot5x-3xy\cdot7y\)
=3xy(3xy+5x-7y)
\(x^2y^2z+xy^2z^2+x^2yz^2=xyz\left(xy+yz+xz\right)\)
2x(x+1)+2(x+1)
=(x+1)(2x+2)
=2(x+1)(x+1)=2(x+1)^2
