Too many words.
Signal amplification is a fact. Negative power is an attempt to clarify a concept of stray reactances leading to a mathematical solution known as a "power factor" - not a fact.
"P:ower" refers to the ability to perform work when force(pressure/voltage) is combined with the capacity to deliver "energy" (electrical current. or current of water or flow of air). In electrical terms Power is defined as
volts x amps.
When dealng with alternating-currents the definitions must be modified
instantaneous power = instantaneoas voltage x instantaneous current
but the presence of (stray) reactances causes the waveforms of voltage and current to lose synchrony (they develop a difference in relative phase). Now multiply values of instantaneous voltag and current, plot them as a graph and you have a sinewave
note that this is a graphical representation of a mathematical function and has no entity as a waveFORM.
The relative phase-shift is referred to as a "power factor" which expresses the ratio between "DC-power" and "AC-power" and it pleases some to regard the lower half-cycles of that graph as representing "negative power"..
In truth the concept can be useful especially when makimg measurements with an ac-bridge. Be careful not to fall into the easy trap of forgetting that you are desling with "short speak".
To cvcalculate he gain provided by an amplifier requires a venture into Algebra and the technique varies depending on thr amplifier concerned is a voltage-Amp, a current-amp or a power-amp.
Follow through the algebraic procedure and find that you are confronted with parts of the circuit where increasing (say) the voltage causes a decrease in current ... a negative value for the circuit resistance. But that is only in the world of mathematics.