Find the roots of the polynomial in $Z_5$

Question

The polynomial is: $2x^{219} + 3x^{74} + 2x^{57} + 3x^{44}$. Find the zeros.

Now my first step, which I believe shall be correct is to reduce the exponents of the polynomial in mod 5.

Thus: 219 becomes 4, 74 becomes 4, 57 becomes 3, and 44 becomes 4.

And our new polynomial ends up simplyfying to $8x^{4} + 2x^{2}$.

Is this getting me on the right track, because doing these calculations in modulus is pretty tricky.

Is there a more simple theorem that I can use to figure this out easier?

Edit: My attempt

I'm assuming i'm on the right track. So I simplify the whole thing down to $2x^3 + 3x^2 + 2x + 3 = 0$

I factor that into $(x^2 + 1)(2x + 1) = 0$

And get roots $i, -i, and -1/2$. Is my logic correct?

Answer 1

No, it's not correct. $x^5 = x$ in $\mathbb Z_5$, so $x^j = x^k$ if $j \equiv k \mod 4$ (not mod $5$) and $j,k > 0$.