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a)
$\frac{x}{x+1}=\frac{3x}{2x+1}$
$x\left(2x+1\right)=3x\left(x+1\right)$
$2x^2+x-3x^2-3x=0$
$-x^2-2x=0$
$-x\left(x+2\right)=0$
$x=0$
$x+2=0$
$x=-2$
b)
$\frac{2x}{1+x}=\frac{x}{1-x}$
$2x\left(1-x\right)=x\left(1+x\right)$
$2x-2x^2-x-x^2$
$-3x^2+x=0$
$x\left(-3x+1\right)=0$
$x=0$
$-3x+1=0$
$3x=1$
$x=\frac{1}{3}$
c)
$\frac{x}{x+1}=\frac{x+2}{x}$
$x^2=\left(x+1\right)\left(x+2\right)$
$x^2=x^2+2x+x+2$
$x^2-x^2-3x-2$
$-3x-2=0$
$x+\frac{2}{3}=0$
$x=-\frac{2}{3}$