a.
$14x^2y-21xy^2+28x^2y^2=7xy(2x-3y+4xy)$
b.
$x(x+y)-5x-5y=x(x+y)-5(x+y)=(x+y)(x-5)$
c.
$10x(x-y)-8(y-x)=10x(x-y)+8(x-y)=(10x+8)(x-y)=2(5x+4)(x-y)$
d.
$(3x+1)^2-(x+1)^2=[(3x+1)-(x+1)][(3x+1)+(x+1)]=2x(4x+2)=4x(2x+1)$
e.
$x^3+y^3+z^3-3xyz=(x+y)^3-3xy(x+y)+z^3-3xyz$
$=(x+y)^3+z^3-[3xy(x+y)+3xyz]=(x+y+z)[(x+y)^2-z(x+y)+z^2]-3xy(x+y+z)$
$=(x+y+z)[(x+y)^2-z(x+y)+z^2-3xy]$
$=(x+y+z)(x^2+y^2+z^2-xy-yz-xz)$
f.
$5x^2-10xy+5y^2-20z^2=5(x^2-2xy+y^2)-20z^2$
$=5(x-y)^2-20z^2=5[(x-y)^2-(2z)^2]=5(x-y-2z)(x-y+2z)$
g.
$x^3-x+3x^2y+3xy^2+y^3-y=(x^3+y^3)-(x+y)+(3x^2y+3xy^2)$
$=(x+y)(x^2-xy+y^2)-(x+y)+3xy(x+y)=(x+y)(x^2-xy+y^2-1+3xy)$
$=(x+y)(x^2+2xy+y^2-1)=(x+y)[(x+y)^2-1^2]$
$=(x+y)(x+y-1)(x+y+1)$
h.
$x^2+7x-8=(x^2-x)+(8x-8)=x(x-1)+8(x-1)=(x-1)(x+8)$
i.
$x^2+4x+3=(x^2+x)+(3x+3)=x(x+1)+3(x+1)=(x+1)(x+3)$
j.
$16x-5x^2-3=(15x-5x^2)+(x-3)=-5x(x-3)+(x-3)=(x-3)(-5x+1)$
k.
$x^4+4=(x^2)^2+2^2+2.2.x^2-4x^2=(x^2+2)^2-(2x)^2$
$=(x^2+2-2x)(x^2+2+2x)$
l.
$x^3-2x^2+x-xy^2=x(x^2-2x+1-y^2)=x[(x-1)^2-y^2]=x(x-1-y)(x-1+y)$
