In order to simplify the expression $\frac{a^{3x}-a^{-3x}}{a^{x}-a^{-x}}$, the numerator can be factorised into $\left(a^{x}-a^{-x}\right)\left(a^{2x}+1+a^{-2x}\right)$.
Similarly, $x^\frac{3}{2}+x^{-\frac{3}{2}}$ can be factorised into $\left(x^\frac{1}{2}+x^{-\frac{1}{2}}\right)\left(x-1+x^{-1}\right)$.
Is there any way I can factorise an expression like these by hand?
$$x^{2n+1}-y^{2n+1}=(x-y)(x^{2n}+x^{2n-1}y+x^{2n-2}y^2+\cdots+xy^{2n-1}+y^{2n})$$ $$x^{2n+1}+y^{2n+1}=(x+y)(x^{2n}-x^{2n-1}y+x^{2n-2}y^2+\cdots-xy^{2n-1}+y^{2n})$$