I presume that you are required here to use the "difference quotient" definition. That is that the derivative is defined as $\displaystyle \lim_{dx\to 0} \frac{P(t+ dt)- P(t)}{dx}$ (Many people do not like using "dt" here, reserving that for after the limit). Yes, with $\displaystyle P(t)= 20t+ 45t^2$, $\displaystyle P(t+ dt)= 20(t+ dt)+ 45(t+ dt)^2$.
But then you have an error in your very next line. You have 20(t+ dt) equal to 20t+ dt. It should be 20t+ 20dt. So $\displaystyle P(t+ dt)= 20t+ 20dt+ 45t^2+ 90t dt+ dt^2$
And subtracting $\displaystyle P(t)= 20t+ 45t^2$ from that leaves $\displaystyle P(t+ dt)- P(t)= 20dt+ 90 t dt+ dt^2$. You have only "dt" rather than "20dt". Finally, dividing by "dt", $\displaystyle \frac{P(t+dt)- P(t)}{dt}= 20+ 90t+ dt$ and the limit of that as dt goes to 0 is 20+ 90t.
As your Q. stands, there is not a specific & stated requirement to use the "difference quotient" definition. Hence, the solution provided in post #2 is adequate. Although the solution given in post #3 is mathematically valid, it is based on an unjustified presumption and is akin to using a sledge hammer to crack a nut.
Al.