$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 5 } } { 4 } \\ x = \dfrac { - 1 - \sqrt{ 5 } } { 4 } \end{array}$
$\left ( \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 5$
$ $ Organize the expression $ $
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 5$
$16 x ^ { 2 } + 8 x + 1 = \color{#FF6800}{ 5 }$
$ $ Move the expression to the left side and change the symbol $ $
$16 x ^ { 2 } + 8 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$16 x ^ { 2 } + 8 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$ $ Subtract $ 5 $ from $ 1$
$16 x ^ { 2 } + 8 x \color{#FF6800}{ - } \color{#FF6800}{ 4 } = 0$
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ 320 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } }$
$x = \dfrac { - 8 \pm \sqrt{ \color{#FF6800}{ 320 } } } { 2 \times 16 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 8 \pm \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { 2 \times 16 }$
$x = \dfrac { - 8 \pm 8 \sqrt{ 5 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } }$
$ $ Multiply $ 2 $ and $ 16$
$x = \dfrac { - 8 \pm 8 \sqrt{ 5 } } { \color{#FF6800}{ 32 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \\ x = \dfrac { - 8 - 8 \sqrt{ 5 } } { 32 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 4 } } } \\ x = \dfrac { - 8 - 8 \sqrt{ 5 } } { 32 } \end{array}$
$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 5 } } { 4 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 5 } } { 4 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 4 } } } \end{array}$