Only one answer

$\sqrt{x+12}=1+\sqrt{2x+1}$

$x+12=(1+\sqrt{2x+1})^2$

$x+12=2+2\sqrt{2x+1}+2x$

$-2\sqrt{2x+1}=-10+x$

$4(2x+1)=(-10+x)^2$

$x^2-28x+96=0$

$x=\frac{28\pm \sqrt{400}}{2}=14\pm10$

Since for $x=24$ : $\sqrt{36}-\sqrt{49}=-1$ , the only solution for the equation is: $\boxed{x=4}$ .