\(\sqrt{12}=\sqrt{x^2+12x+13}\)
\(\Leftrightarrow12=x^2+12x+13\)
\(\Leftrightarrow x^2+12x+36-35=0\)
\(\Leftrightarrow\left(x+6\right)^2-35=0\)
\(\Leftrightarrow\left(x+6+\sqrt{35}\right)\left(x+6-\sqrt{35}\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=-6-\sqrt{35}\\x=\sqrt{35}-6\end{cases}}\)