Appendix C - Induction and the Binomial Theorem - Exercises - Page A15: 15

$x^4+4x^2+6+4x^{-2}+x^{-4}$

We need to expand $(x+x^{-1})^4$ by using Pascal`s Triangle. Therefore, we have: $(x+x^{-1})^4=1x^4(x^{-1})^0+4x^3(x^{-1})^1+6x^2(x^{-1})^2+4x^1(x^{-1})^3+1x^0(x^{-1})^4\\=x^4+4x^3x^{-1}+6x^2x^{-2}+4xx^{-3}+x^{-4} \\=x^4+4x^2+6+4x^{-2}+x^{-4}$