We want to write [tex]n^2+8n+59[/tex] in the form [tex](n-a)^2+b[/tex]. Expand the second expression and set it equal to the first:
[tex]n^2-2an+a^2+b=n^2+8n+59[/tex]
So we need to have
[tex]\begin{cases}-2a=8\\a^2+b=59\end{cases}\implies a=-4,b=43[/tex]
Then
[tex]n^2+8n+59=(n+4)^2+43=0\implies(n+4)^2=-43\implies n+4=\pm i\sqrt{43}[/tex]
[tex]\implies n=-4\pm i\sqrt{43}[/tex]
