Answer
$x_1=-\dfrac{1}{2} - \dfrac{\sqrt{85}}{2}
\\x_2=-\dfrac{1}{2} + \dfrac{\sqrt{85}}{2}$
Work Step by Step
RECALL:
$\log_a{x}=y \longrightarrow a^y=x, a\gt0, a\ne1$
Use the rule above to obtain:
$\log_5{(x^2+x+4)}=2
\\\longrightarrow 5^2=x^2+x+4
\\25=x^2+x+4$
Subtract by 25 on both sides of the equation to obtain:
$25-25=x^2+x+4-25
\\0=x^2+x-21$
Use the quadratic formula, $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a = 1, b=1,$ and $c=-21$ to obtain:
$x=\dfrac{-1 \pm \sqrt{1^2-4(1)(-21)}}{2(1)}
\\x=\dfrac{-1 \pm \sqrt{1-(-84)}}{2}
\\x=\dfrac{-1 \pm \sqrt{1+84}}{2}
\\x=\dfrac{-1 \pm \sqrt{85}}{2}$
Split the solutions to obtain:
$x_1=-\dfrac{1}{2} - \dfrac{\sqrt{85}}{2}
\\x_2=-\dfrac{1}{2} + \dfrac{\sqrt{85}}{2}$