a) \(x^8+x+1\)

\(=x^8+x^7-x^7+x^6-x^6+x^5-x^5+x^4-x^4+x^3-x^3+x^2-x^2+x+1\)

\(=\left(x^8+x^7+x^6\right)-\left(x^7+x^6+x^5\right)+\left(x^5+x^4+x^3\right)-\left(x^4+x^3+x^2\right)+\left(x^2+x+1\right)\)

\(=x^6\left(x^2+x+1\right)-x^5\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-x^2\left(x^2+x+1\right)+\left(x^2+x+1\right)\)

\(=\left(x^2+x+1\right)\left(x^6-x^5+x^3-x^2+1\right)\)

c) \(x^4+4y^4=x^4+4x^2y^2+4y^4-4x^2y^2=\left(x^4+4x^2y^2+4y^4\right)-4x^2y^2\)

\(=\left(x^2+2y^2\right)^2-\left(2xy\right)^2=\left(x^2+2y^2-2xy\right)\left(x^2+2y^2+2xy\right)\)

d) \(x^4+4=x^4+4x^2+4-4x^2=\left[\left(x^2\right)^2+2\cdot x^2\cdot2+2^2\right]-4x^2\)

\(=\left(x^2+2\right)^2-\left(2x\right)^2=\left(x^2+2-2x\right)\left(x^2+2+2x\right)\)

\(=\left(x^2-2x+2\right)\left(x^2+2x+2\right)\)