Aloha :)
$$\left.\lg(x+1)=1+\lg(2x+1)\quad\right|1=\lg(10)$$$$\left.\lg(x+1)=\lg(10)+\lg(2x+1)\quad\right|\lg(a)+\lg(b)=\lg(a\cdot b)$$$$\left.\lg(x+1)=\lg(10\cdot(2x+1))=\lg(20x+10)\quad\right|10^{\cdots}$$$$\left.x+1=20x+10\quad\right|-x$$$$\left.1=19x+10\quad\right|-10$$$$\left.-9=19x\quad\right|:\,19$$$$\left.x=-\frac{9}{19}\approx-0,473684\quad\right.$$