1. ## congruent mod

Determine whether or not x3 +2x+1 and x4 +3x-2 are congruent mod
x2 + 2x + 2 in Q[x]. Answer the same question for x4 + x3 + x2 + 2 and x3 + 1
mod(x2 + 1) in Z3[x].

We are asked whether or not $\displaystyle x^3+2x+1 \equiv x^4+3x-2 \mod{x^2+2x+2}$ is true in the space of rational numbers.
If this were true then $\displaystyle x^2+2x+2$ divides the difference $\displaystyle (x^4+3x-2) - (x^2+2x+1) = x^4-x^2+x-3$ with a rational quotient and no remainder.