$(p + 2)$th term from the end in the binomial expansion of

${\left(x^2-\frac{2}{x^2}\right)}^{2n+1}$ is

$(p + 2)$th term from the end

$=[2n+2-(p+1\left)\right]\mathrm{th}=(2n-p+1)\mathrm{th}$

from the beginning 

and $T_{2n-p+1}{=}^{2n+1}{C}_{2n-p}{\left(x^2\right)}^{2n+1-(2n-p)}{\left(-\frac{2}{x^2}\right)}^{2n-p}$

${=}^{2n+1}{C}_{2n-p}(-2{)}^{2n-p}{x}^{2p+1-2n}$