a: \(x^4y^4+64\)
\(=x^4y^4+16x^2y^2+64-16x^2y^2\)
\(=\left(x^2y^2+8\right)^2-\left(4xy\right)^2\)
\(=\left(x^2y^2-4xy+8\right)\left(x^2y^2+4xy+8\right)\)
b: \(x^3+3x^2y+3xy^2+y^3-x-y\)
\(=\left(x+y\right)^3-\left(x+y\right)\)
\(=\left(x+y\right)\left[\left(x+y\right)^2-1\right]\)
\(=\left(x+y\right)\left(x+y+1\right)\left(x+y-1\right)\)
c: \(x^8+x+1\)
\(=x^8+x^7+x^6-x^7-x^6-x^5+x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1\)
\(=x^6\left(x^2+x+1\right)-x^5\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-x^2\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^6-x^5+x^3-x^2+1\right)\)
d: \(x^3+y^3-2\left(x^2-y^2\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)-2\left(x-y\right)\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2-2x+2y\right)\)
e: \(\left(x^2+y^2\right)-2x^2-2y+1\)
\(=\left(y^2-2y+1\right)-x^2\)
\(=\left(y-1\right)^2-x^2\)
\(=\left(y-1-x\right)\left(y-1+x\right)\)
f: \(x^2+5x+6\)
\(=x^2+2x+3x+6\)
=x(x+2)+3(x+2)
=(x+2)(x+3)
g: \(\left(x^2+x+1\right)\left(x^2+x+2\right)-6\)
=\(\left(x^2+x\right)^2+3\left(x^2+x\right)+2-6\)
\(=\left(x^2+x\right)^2+3\left(x^2+x\right)-4\)
\(=\left(x^2+x+4\right)\left(x^2+x-1\right)\)
