How was this equation derived for a L-C negative resistance oscillator?
The original has V as a common factor. If we assume there can be some other V than zero, the V can be divided off. Then multiple both sides with sL+r and reorder the terms. That's a formula which shows the possible s. If the solved s happens to be complex and have positive real part the oscillator really can oscillate. The limit is imaginary s, then the amplitude stays and the oscillation is pure sinusoidal.