Find the value of the prime $p$ knowing that $p>5$ and $p│(3^{p+1} +5^{p-1} + 1)$

Question

I am trying to find the value of $p$, a prime number of which we only know that $p>5$ and:

$p│(3^{p+1} +5^{p-1} + 1)$

This is part of a collection of exercises regarding divisibility, Fermat's little theorem and congruences (among others).

Seeing that the second term has $5^{p-1}$, I imagine Fermat's little theorem could be used on that, but I don't know how to advance from there.

Any help/hints are welcome!

(currently stuck trying to take the value from the main expression knowing that $3^{p-1}\equiv 1\bmod p$ and $3^{p-1}*3^2\equiv 3^2\bmod p$, so $3^{p+1}\equiv 3^2\bmod p$, also $5^{p-1}\equiv 1\bmod p$)

Answer 1

Hint: If $p│3^{p+1}+5^{p-1}+1$, that is the same as saying that $3^{p+1}+5^{p-1}+1\equiv 0\bmod p$. But you know the values of $3^{p+1}\bmod p$ and $5^{p-1}\bmod p$.