#### Answer

$x=-1$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\sqrt{2x+3}=x+2
,$ square both sides and then express in the form $ax^2+bx+c=0.$ Use concepts of solving quadratic equations to find the values of $x$. Finally, do checking of the solutions with the original equation.
$\bf{\text{Solution Details:}}$
Squaring both sides of the given equation results to
\begin{array}{l}\require{cancel}
2x+3=(x+2)^2
.\end{array}
Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
2x+3=(x)^2+2(x)(2)+(2)^2
\\\\
2x+3=x^2+4x+4
.\end{array}
In the form $ax^2+bx+c=0,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
-x^2+(2x-4x)+(3-4)=0
\\\\
-x^2-2x-1=0
\\\\
-1(-x^2-2x-1)=-1(0)
\\\\
x^2+2x+1=0
.\end{array}
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the factored form of the equation above is
\begin{array}{l}\require{cancel}
(x+1)(x+1)=0
\\\\
(x+1)^2=0
.\end{array}
Taking the square root of both sides results to
\begin{array}{l}\require{cancel}
x+1=\pm\sqrt{0}
\\\\
x+1=0
\\\\
x=-1
.\end{array}
Upon checking, $
x=-1
$ satisfies the original equation.