Define $f(x) = x^6 + x^5 + 3x^4 +x^3 + 3x^2 + x + 1$. Find the largest prime factor of $f(19) + 1$ (Homework)

Question

Define $f(x) = x^6 + x^5 + 3x^4 +x^3 + 3x^2 + x + 1$. Find the largest prime factor of $f(19) + 1$ This problem is from a homework set of my class at source: Alphastar.academy. I believe there a number of ways to factor this and solve it, and I would appreciate it if I were able to see a couple methods on how to do this problem.

Answer 1

Conveniently, $f(x)+1$ factors as $$(x^2-x+1)(x^2+x+1)(x^2+x+2)$$ With $x=19$ this produces the three factors $343×381×382$, from which we work out that the largest prime factor is $191$ (of $382$).