a) \(2x-6y\)
\(=2\left(x-3y\right)\)
b) \(x^2-4xy+4y^2\)
\(=x^2-2.x.2y+\left(2y\right)^2\)
\(=\left(x-2y\right)^2\)
c) \(x^2+2x-y^2+1\)
\(=\left(x^2+2x+1\right)-y^2\)
\(=\left(x+1\right)^2-y^2\)
\(=\left(x+1-y\right)\left(x+1+y\right)\)
d) \(x^2+2xy-y^2+1\)
\(=-\left(x^2-2xy+y^2-1\right)\)
\(=-\left[\left(x-y\right)^2-1\right]\)
\(=-\left(x-y-1\right)\left(x-y+1\right)\)
e) \(x^3+y^3+z^3-3xyz\)
\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)
\(=\left[\left(x+y\right)^3+z^3\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)^3-3\left(x+y+z\right)\left(x+y\right)z-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)^3-3\left(x+y+z\right)\left(xy+xz+yz\right)\)
\(=\left(x+y+z\right)\left[\left(x+y+z\right)^2-3xy-3xz-3yz\right]\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2+2xy+2xz+2yz-3xy-3xz-3yz\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)\)
f) \(x^4+y^4\)
\(=\left(x^2\right)^2+2x^2y^2+\left(y^2\right)^2-2x^2y^2\)
\(=\left(x^2+y^2\right)^2-\left(\sqrt{2}xy\right)^2\)
\(=\left(x^2+y^2-\sqrt{2}xy\right)\left(x^2+y^2+\sqrt{2}xy\right)\)
