${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{x_n}& =(n+1{)}^2+(n+2{)}^2+[(n+1)(n+2{)]}^2\\ \multicolumn{1}{c}{x_n}& =n^2+2n+1+n^2+4n+4+[n^2+3n+2{]}^2\\ \multicolumn{1}{c}{}& =n^4+6n^3+15n^2+18n+9\\ \multicolumn{1}{c}{}& =[n^2+3n+3{]}^2\end{array}}$