$\begin{array}{l}
Gia\,su:{n^2} + 2n + 202 = {m^2} \Leftrightarrow {\left( {n + 1} \right)^2} + 201 = {m^2}\\
\Leftrightarrow 201 = {m^2} - {\left( {n + 1} \right)^2} = \left( {m + n + 1} \right)\left( {m - n - 1} \right)\\
Ma\,201 = 1.201 = \left( { - 1} \right).\left( { - 201} \right) = 3.67 = \left( { - 3} \right).\left( { - 67} \right)\,nen:\\
TH1:\left\{ \begin{array}{l}
m + n + 1 = 1\\
m - n - 1 = 201
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = 101\\
n = - 101
\end{array} \right.\\
TH2:\left\{ \begin{array}{l}
m + n + 1 = - 1\\
m - n - 1 = - 201
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = - 101\\
n = 99
\end{array} \right.\\
TH3:\left\{ \begin{array}{l}
m + n + 1 = 3\\
m - n - 1 = 67
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = 35\\
n = - 33
\end{array} \right.\\
TH4:\left\{ \begin{array}{l}
m + n + 1 = - 3\\
m - n - 1 = - 67
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = - 35\\
n = 31
\end{array} \right.\\
TH5:\left\{ \begin{array}{l}
m + n + 1 = - 67\\
m - n - 1 = - 3
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = - 35\\
n = - 33
\end{array} \right.\\
TH6:\left\{ \begin{array}{l}
m + n + 1 = 67\\
m - n - 1 = 3
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = 35\\
n = 31
\end{array} \right.\\
TH7:\left\{ \begin{array}{l}
m + n + 1 = - 201\\
m - n - 1 = - 1
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = - 101\\
n = - 101
\end{array} \right.\\
TH8:\left\{ \begin{array}{l}
m + n + 1 = 201\\
m - n - 1 = 1
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = 101\\
n = 99
\end{array} \right.\\
Vay\,n \in \left\{ { - 101;99; - 33;31} \right\}
\end{array}$
