$\text{I.}2x-\left(x+3\right)=6+2x$  

$2x-x-3=6+2x$  

$x-3=6+2x\mid +3$  

$x=9+2x\mid -2x$  

$-x=9\mid \cdot \left(-1\right)$  

$x=-9$  

 

$\text{II.}\frac{1}{2}\left(x+5\right)=\frac{2}{3}x+1\mid \cdot 6$  

${\sout{6}}^3\cdot \frac{1}{{\sout{2}}^1}\cdot \left(x+5\right)={\sout{6}}^2\cdot \frac{2}{{\sout{3}}^1}x+6\cdot 1$   

$3\left(x+5\right)=4x+6$       

$3x+15=4x+6\mid -3x$  

$15=x+6\mid -6$  

$x=9$  

 

$\text{III.}\frac{x+1}{2}-\frac{x}{5}=\frac{5x+2}{10}\mid \cdot 10$  

${\sout{10}}^5\cdot \frac{x+1}{{\sout{2}}^1}-{\sout{10}}^2\cdot \frac{x}{{\sout{5}}^1}={\sout{10}}^1\cdot \frac{5x+2}{{\sout{10}}^1}$  

$5\left(x+1\right)-2x=5x+2$  

$5x+5-2x=5x+2\mid -5x$  

$5-2x=2\mid -5$  

$-2x=-3\mid :\left(-2\right)$  

$x=\frac{3}{2}$  

 

$\text{IV}x\left(1+x\right)-1=-x^2+2\left(3x+x^2\right)$  

$x+x^2-1=-x^2+6x+2x^2$  

$x+x^2-1=x^2+6x\mid -x^2$  

$x-1=6x\mid -x$  

$-1=5x\mid :5$  

$x=-\frac{1}{5}$