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**1. Homework Statement**

Find all p, prime for which x+2 is a factor of f(x) = 5x

^{4}- 2x

^{3}+ 3x

^{2}+ 4x - 1 in Z

_{p}

**2. Homework Equations**

**3. The Attempt at a Solution**

So in Z

_{p}, x = p-2

I tried the first 4 primes and got the following results:

Z

_{3}, x=1, f(x) = 9 = 0

Z

_{5}, x=3, f(x) = 390 - 1 =/ 0

Z

_{7}, x=5, f(x) = 2970 - 1 =/ 0

Z

_{11}, x=9, f(x) = 31626 - 1 =/ 0

Then the only way x+2 is a factor of f(x) is if p | f(x)

However, f(x) = k(p-2) - 1 for all prime numbers, and for all k natural numbers

Then since f(x) is some multiple of p-2, -1, p never divides f(x) for all primes >= 5.

Do I have the right idea? How can I formulate this so it is more formal since it's quite sloppy..

thanks!! :)