How many natural value of n such that $n^5+2n^4+n-1$ is prime number?

Question

From above polynomial, I can only get one value to make it prime.

The value, I guess, is only one.

For $n=1$, we got:

$$(n^5+2n^4+n-1)= 1+2+1-1= 3 \quad\text{(prime)}$$

But, I cannot find the other value of $n$, such that the polynomial is prime.

My question: How to prove that the solution is trivial?

Thanks

Answer 1

Hint:

$$ n^5+2n^4+n-1 = (n^2+n-1) (n^3+n^2+1)$$