#### Answer

$x=\left\{ \dfrac{-1-2\sqrt{5}}{4},\dfrac{-1+2\sqrt{5}}{4} \right\}
$

#### Work Step by Step

Taking the square root of both sides, the solutions to the given equation, $
(4x+1)^2=20
,$ are
\begin{array}{l}\require{cancel}
4x+1=\pm\sqrt{20}
\\\\
4x+1=\pm\sqrt{4\cdot5}
\\\\
4x+1=\pm\sqrt{(2)^2\cdot5}
\\\\
4x+1=\pm2\sqrt{5}
\\\\
4x=-1\pm2\sqrt{5}
\\\\
x=\dfrac{-1\pm2\sqrt{5}}{4}
.\end{array}
Hence, the solutions are $
x=\left\{ \dfrac{-1-2\sqrt{5}}{4},\dfrac{-1+2\sqrt{5}}{4} \right\}
.$