#### Answer

$x=-1$

#### Work Step by Step

Squaring both sides of the equation and then using the properties of equality, we obtain:
\begin{array}{l}\require{cancel}\left(
\sqrt{x^2+x+4}
\right)^2=\left(
x+3
\right)^2
\\\\
x^2+x+4=(x)^2+2(x)(3)+(3)^2
\\\\
x^2+x+4=x^2+6x+9
\\\\
(x^2-x^2)+(x-6x)=9-4
\\\\
-5x=5
\\\\
x=\dfrac{5}{-5}
\\\\
x=-1
.\end{array}
Upon checking, $
x=-1
$ satisfies the original equation.