I'm a little bored, today, so I'll launch into a short lecture.
This is a non-trivial task, in general. It's quite a bit easier using additional information from the Calculus, but since you posted in "Algebra", we are limited and we may need some hope, rather than an obvious pathway..
Standard Form: P^5 + 4P^4 - C/S = 0
We may need to HOPE there is a solution on 0 < P < 1. The task is to PROVE it. As the degree is 5 (odd) there must be a solution, somewhere, but other guarantees will have to place it in the right interval.
Let's investigate with the Intermediate Value Theorem! (IVT)
The idea is that if we can find some value less than zero and another value greater than zero, then we must hit zero somewhere between.
Clearly, for P = 0, we have 0^5 + 4(0^4) - C/S = -C/S < 0. That's part of it.
Also, for P = 1, we have 1^5 + 4(1^4) - C/S = 1 + 4 - C/S = 5 - C/S, and I can't tell if that's positive, or not. If it is, we're in luck! So, C/S better be less than 5. Since C < S, we have C/S < 1 < 5 and 5 - C/S > 0.
Thus, there IS a solution on 0 < P < 1, based on the promise of the IVT.
Well, we now have permission to FIND the solution, knowing that there is one. What's your plan?
One way is just to evaluate the expression somewhere in the interval and see what you get.
We have P = 0 ==> Something Negative
We have P = 1 ==> Something Positive
It is common to try P = 1/2. What do you suppose we should do next if that leads to a negative value?
There are faster ways, but the process of this one is wonderfully obvious.
