How do I get the expansion of $(x^2+2)^{−1}$ to the $x^3$ term

Question

How do I get the expansion of $(x^2+2)^{−1}$ to the $x^3$ term

I don't really know how I have to use the formula here. In wich form do I have to put $(x^2+2)^{−1}$?

Do i need to factorise it ?

Answer 1

Outline: Note that $$(x^2+2)^{-1}=\frac{1}{2}\cdot \frac{1}{1+\frac{x^2}{2}}.$$ Then use the expansion $\frac{1}{1-t}=1+t+t^2+t^3+\cdots$, valid for $|t|\lt 1$, with $t=-\frac{x^2}{2}$.