<s>[INST] Complex Roots in a Complex Equation 
Problem: The equation $$\frac{x}{x+1} + \frac{x}{x+2} = kx$$ has exactly two complex roots. Find all possible complex values for $k.$


Progress: I was online today, and I saw this problem. Heres what I tried: $$(x+1)x+(x+2)x=kx(x+1)(x+2),$$ and I immediately got stuck :(. I'm quite new at complex numbers. Help? Thanks.
 [/INST] Good!
Yes, multiplying both sides by $(x+ 1)(x+ 2)$ "reduces" the equation to $(x+ 1)x+ (x+ 2)x= kx(x+ 1)(x+ 2)$.  Now, I would do the indicated multiplications to get $x^2+ x+ x^2+ 2x= kX*(x^2+ 3x+ 2)= kx^3+ 3kx^2+ 2kx$ which can be written as $kx^3+ (3k- 1)x^2+ (2k- 1)x= 0$.  The left side can be factored as $x(kx^2+ (3k- 1)x+ (2k- 1))= 0$.  One root is $x= 0$ and, if $x$ is not $0$, $kx^2+ (3k- 1)x+ (2k-1)= 0$ can be solved by the "quadratic formula".
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