\(\sqrt{x^2+5x+20}=4\)
\(\Leftrightarrow\) \(x^2+5x+20=16\)
\(\Leftrightarrow\) \(x^2+5x+4=0\)
\(\Leftrightarrow\) \(\left(x^2+x\right)+\left(4x+4\right)=0\)
\(\Leftrightarrow\) \(x\left(x+1\right)+4\left(x+1\right)=0\)
\(\Leftrightarrow\) \(\left(x+1\right).\left(x+4\right)=0\)
\(\Leftrightarrow\) \(\left[{}\begin{matrix}x+1=0\\x+4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=-4\end{matrix}\right.\)