Answer
$x=\{ 2,6 \}$
Work Step by Step
Squaring both sides of the given equation, $
\sqrt{x-2}+3=\sqrt{4x+1}
,$ results to
\begin{array}{l}\require{cancel}
(\sqrt{x-2}+3)^2=(\sqrt{4x+1})^2
\\
(\sqrt{x-2})^2+2(\sqrt{x-2})(3)+(3)^2=4x+1
\\
x-2+6\sqrt{x-2}+9=4x+1
\\
6\sqrt{x-2}=(4x-x)+(1+2-9)
\\
6\sqrt{x-2}=3x-6
\\
\dfrac{6\sqrt{x-2}}{3}=\dfrac{3x-6}{3}
\\
2\sqrt{x-2}=x-2
.\end{array}
Squaring both sides for the second time results to
\begin{array}{l}\require{cancel}
(2\sqrt{x-2})^2=(x-2)^2
\\
4(x-2)=(x)^2-2(x)(2)+(2)^2
\\
4x-8=x^2-4x+4
\\
0=x^2+(-4x-4x)+(4+8)
\\
x^2-8x+12=0
\\
(x-6)(x-2)=0
.\end{array}
Equating each factor to zero (Zero Product Property) and then solving for the variable, the solutions are $
x=\{ 2,6 \}
.$
Upon checking, both solutions, $
x=\{ 2,6 \}
,$ satisfy the original equation.