Answer
$\left\{\dfrac{-1+\sqrt{85}}{2}, \dfrac{-1-\sqrt{85}}{2}\right\}$
Work Step by Step
$\because y=\log_a x \text{ is equivalent to } x= a^y$
$\therefore 2 = \log_5 {x^2+x+4} \text{ is equivalent to } x^2+x+4=5^2$
Solve teh equation above to obtain:
\begin{align*}
x^2+x+4&=5^2\\\\
x^2+x+4&=25\\\\
x^2+4+4-25&=0\\\\
x^2+x-21&=0
\end{align*}
Comparing $x^2+x-21=0$ to $ax^2+bx+c=0$ to find $a,b \text{ and } c$
$$\therefore a =1, b=1, c =-21$$
Evaluating the discriminant $b^2-4ac$ gives:
$$b^2-4ac = (1)^2-4 \times 1\times -21 = 85$$
Solve using the quadratic formula to obtain:
$$x= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$$
$$x= \dfrac{-1\pm \sqrt{85}}{2\times 1}$$
$$x=\dfrac{-1\pm \sqrt{85}}{2}$$
$\therefore x =\dfrac{-1+\sqrt{85}}{2}\hspace{20pt} \text{or} \hspace{20pt} x=\dfrac{-1-\sqrt{85}}{2}$
