What value of x makes the three terms x, x/(x + 1) and 3x/[(x + 1)(x + 2)] those of a geometric sequence?

For the terms x, x/(x + 1) and 3x/[(x + 1)(x + 2)] to be
the terms of a geometric sequence, than the following identity must be
valid:

x^2/(x+1)^2 = 3x^2/[(x + 1)(x +
2)]

We'll multiply by x+1 both sides: x^2/(x+1) =
3x^2/(x+2)

We'll cross multiply: 3x^2*(x+1) = x^2*(x+2)
3x^3 + 3x^2=x^3+2x^2

We'll shift all terms to the left
side: 3x^3 + 3x^2-x^3-2x^2 = 0 2x^3 + x^2 = 0

We'll
factorize by x^2: x^2(2x+1) = 0

We'll cancel the 1st
factor: x^2=0 x1=x2=0

We'll cancel the 2nd factor: 2x+1=0
2x=-1 x=-1/2

The values of x that makes the
three terms x, x/(x + 1) and 3x/[(x + 1)(x + 2)] those of a geometric sequence are:
{-1/2 , 0}.