Answer
$x=-\dfrac{1}{2}$
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+2x+3}
\right)^2=\left(
x+2
\right)^2
\\\\
x^2+2x+3=(x)^2+2(x)(2)+(2)^2
\\\\
x^2+2x+3=x^2+4x+4
\\\\
(x^2-x^2)+(2x-4x)=4-3
\\\\
-2x=1
\\\\
x=\dfrac{1}{-2}
\\\\
x=-\dfrac{1}{2}
.\end{array}
Upon checking, $
x=-\dfrac{1}{2}
$ satisfies the original equation.