Answer
$x=\{ 0,4 \}$
Work Step by Step
Squaring both sides of the given equation, $
\sqrt{3x+4}-1=\sqrt{2x+1}
,$ results to
\begin{array}{l}\require{cancel}
(\sqrt{3x+4}-1)^2=(\sqrt{2x+1})^2
\\
(\sqrt{3x+4})^2-2(\sqrt{3x+4})(1)+(1)^2=2x+1
\\
3x+4-2\sqrt{3x+4}+1=2x+1
\\
(3x-2x)+(4+1-1)=2\sqrt{3x+4}
\\
x+4=2\sqrt{3x+4}
.\end{array}
Squaring both sides for the second time results to
\begin{array}{l}\require{cancel}
(x+4)^2=(2\sqrt{3x+4})^2
\\
(x)^2+2(x)(4)+(4)^2=4(3x+4)
\\
x^2+8x+16=12x+16
\\
x^2+(8x-12x)+16-16=0
\\
x^2-4x=0
\\
x(x-4)=0
.\end{array}
Equating each factor to zero (Zero Product Property) and then solving for the variable, the solutions are $
x=\{ 0,4 \}
.$
Upon checking, both solutions, $
x=\{ 0,4 \}
,$ satisfy the original equation.