\(\left(x+1\right)^4+\left(x-1\right)^4\)
\(=\left(x+1\right)^2.\left(x+1\right)^2+\left(x-1\right)^2.\left(x-1\right)^2\)
\(=\left(x^2+2x+1\right).\left(x^2+2x+1\right)+\left(x^2-2x+1\right).\left(x^2-2x+1\right)\)
\(=x^4+2x^3+x^2+2x^3+4x^2+2x+x^2+2x+1+x^4-2x^3+x^2-2x^3+4x^2-2x+x^2-2x+1\)
\(=2x^4+12x^2+2\)