$$\eqalign{
& {\left( {{n^2} - 3} \right)^2} + 16 = {k^2}\,\,\left( {k \in Z} \right) \cr
& \Leftrightarrow {k^2} - {\left( {{n^2} - 3} \right)^2} = 16 \cr
& \Leftrightarrow \left( {k - {n^2} + 3} \right)\left( {k + {n^2} - 3} \right) = 16 \cr
& Do\,\,k,\,\,n \in Z \Rightarrow k - {n^2} + 3 \in U\left( {16} \right) = \left\{ { \pm 1; \pm 2; \pm 4; \pm 8; \pm 16} \right\} \cr
& TH1:\,\,\left\{ \matrix{
k - {n^2} + 3 = 1 \hfill \cr
k + {n^2} - 3 = 16 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = 8 \hfill \cr
{n^2} = 10\,\,\left( {Loai} \right) \hfill \cr} \right. \cr
& TH2:\,\,\left\{ \matrix{
k - {n^2} + 3 = - 1 \hfill \cr
k + {n^2} - 3 = - 16 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = - {{17} \over 2} \hfill \cr
{n^2} = - {9 \over 2}\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
& TH3:\,\,\left\{ \matrix{
k - {n^2} + 3 = 2 \hfill \cr
k + {n^2} - 3 = 8 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = 5 \hfill \cr
{n^2} = 6\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
& TH4:\,\,\left\{ \matrix{
k - {n^2} + 3 = - 2 \hfill \cr
k + {n^2} - 3 = - 8 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = - 5 \hfill \cr
{n^2} = 0\, \Leftrightarrow n = 0 \hfill \cr} \right.\,\,\left( {tm} \right)\, \cr
& TH5:\,\,\left\{ \matrix{
k - {n^2} + 3 = 4 \hfill \cr
k + {n^2} - 3 = - 4 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = 0 \hfill \cr
{n^2} = - 1\, \hfill \cr} \right.\,\,\left( {Loai} \right)\, \cr
& TH6:\,\,\left\{ \matrix{
k - {n^2} + 3 = - 4 \hfill \cr
k + {n^2} - 3 = 4 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = 0 \hfill \cr
{n^2} = 7 \hfill \cr} \right.\,\,\left( {Loai} \right)\, \cr
& TH7:\,\,\left\{ \matrix{
k - {n^2} + 3 = 8 \hfill \cr
k + {n^2} - 3 = 2 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = 5 \hfill \cr
{n^2} = 0 \Leftrightarrow n = 0\, \hfill \cr} \right.\,\,\left( {tm} \right) \cr
& TH8:\,\,\left\{ \matrix{
k - {n^2} + 3 = - 8 \hfill \cr
k + {n^2} - 3 = - 2 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = - 5 \hfill \cr
{n^2} = 6\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
& TH9:\,\,\left\{ \matrix{
k - {n^2} + 3 = 16 \hfill \cr
k + {n^2} - 3 = 1 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = {{17} \over 2} \hfill \cr
{n^2} = - {9 \over 2}\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
& TH10:\,\,\left\{ \matrix{
k - {n^2} + 3 = - 16 \hfill \cr
k + {n^2} - 3 = - 1 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
k = - {{17} \over 2} \hfill \cr
{n^2} = {{21} \over 2}\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
& Vay\,\,n = 0 \cr} $$
