Just to make sure I get it right.

$\displaystyle \mathbb F_5[x]/(x^2+2) $ is a field.

We want to express $\displaystyle (x^2 + x) \cdot (x^3 + 1) - (x + 1)$ as a polynomial of degree less than 2.

$\displaystyle (x^2+2-2+x)\cdot(x^3+2 -2x+1)-(x + 1)$

$\displaystyle (x-2) \cdot (1-2x) - (x + 1)$

$\displaystyle 4x - 2x^2-3$

$\displaystyle 4x - 2x^2-4+1$

$\displaystyle 4x +1$

Thank you!

Also, I have to find the roots of $\displaystyle t^2+3$

$\displaystyle t=\pm\sqrt{-3}=\pm\sqrt{2}$ but 2 is not a perfect square in $\displaystyle \mathbb Z/5\mathbb Z$

$\displaystyle t^2+3$ has no root.

Is that right????