Answer
$x=4 $
Work Step by Step
$\sqrt{2x+1}+1=x$
$\sqrt{2x+1}=x-1$
Square the both sides $\Rightarrow(\sqrt{2x+1})^2=(x-1)^2$
Expand $\Rightarrow2x+1=x^2-2x+1$
$\Rightarrow x^2-4x=0$
Factor $x(x-4)=0$
$x=0$ or $x-4=0\Rightarrow x=4$
Check the answer:
$x=0$
LHS $\sqrt{2(0)+1}+1=\sqrt{1}+1=1+1=2$
RHS $0$
LHS $\ne$ RHS
$x=0 \color{red}{reject}$
$x=4$
LHS $\sqrt{2(4)+1}+1=\sqrt{8+1}+1=\sqrt{9}+1=3+1=4$
RHS $4$
LHS $=$ RHS
$x=4 $