Can someone explain to me why 25t^3 - 20t^2 + 4t is prime?

Each term has a t in common, so we can factor that out

25t^3 - 20t^2 + 4t = t(25t^2-20t+4)

The fact that we can factor out something that isn't 1 means that the original expression is not prime.

As for 25t^2-20t+4, it can be factored to (5t-2)^2 or (5t-2)(5t-2) following the steps below

25t^2 - 20t + 4

25t^2 -10t - 10t + 4 ... see note below

(25t^2-10t) + (-10t + 4)

5t(5t - 2) - 2(5t - 2)

(5t-2)(5t-2)

(5t-2)^2

note: I used the AC method here. The idea is you multiply the first and last terms (25 and 4) to get 100. Then find two numbers that multiply to 100 and add to -20, which is the middle coefficient term. The two numbers in question are -10 and -10. So that's how I broke up the -20t into -10t-10t. After this step, you factor by grouping.

Overall, the original expression fully factors to t(5t-2)^2

The fact that we can factor like this means it is not prime.

I have a feeling your teacher made a typo somewhere.

see below

Step-by-step explanation:

25t^3 - 20t^2 + 4t

Factor out a t

t(25t^2 -20t+4)

We notice that this is a perfect square trinomial

a^2 -2ab +b^2 where a = 5t  and b=2

This factors into (a-b)^2

t(25t^2 -20t+4) = t(5t-2) ^2

Since this factors, it is not prime

sorry i dont know my friend