we have the equation
[tex]x^2+18x+90=0[/tex]Complete the square
[tex]\begin{gathered} (x^2+18x)+90=0 \\ (x^2+18x+9^2-9^2)+90=0 \\ (x^2+18x+9^2)+90-9^2=0 \\ (x^2+18x+81)+9=0 \end{gathered}[/tex]Rewrite as perfect squares
[tex]\begin{gathered} (x+9)^2+9=0 \\ (x+9)^2=-9 \end{gathered}[/tex]take square root both sides
[tex]\sqrt{(x+9)^2}=\pm\sqrt{-9}[/tex][tex]\begin{gathered} x+9=\pm3i \\ x=-9\pm3i \end{gathered}[/tex]