Asymptotic bode plots
Quick Answer cause I'm at work and don't have time to do the plotting but basically, in equation b you already have the expression i what i like to call "The Bode Form", to only a gain and terms of the form $(1+s/A)$ with $A$ some value. I'm sure the professor explained this, that when you take logarithm you add and subtract all this different terms because the log turns multiplication into addition and division into subtraction, etc. This individual terms, When $s &lt; A$ are approximately equal to 1 (a constant line at 0db Gain in the plot), and when$ s &gt; A$ are approximately equal to $s/A$ ( a 20db/decade drop in the gain).

So the plot would be an increasing line of 20db/decade from  up to your first pole at 20 rad/s, where the gain would be $20\log_{10}(3600)$, and then it would remain at that until 100 rad/s where the next pole appears which again makes the gain drop at 20db/decade, and then at the next pole at 1000 rad/s it drops at 40db/decade and that's it, that's your plot.

Could you maybe put up the image you have in your notebook so everyone could see what the professor did.



Attempt at illustrating below. This video may be instructive.