I need to calculate the product of $(x^2 + 3x + 1)(x^2 + 4x + 3)\pmod{x^3 + 1}$, where the product is in $\mathbb{Z}_5[x]$.
Is this problem as simple as just multiplying the two, which would be $4x^4 + 7x^3 +16x^2 + 9x + 3$, and then dividing $x^3 + 1$ into it?