How to calculate modulo of two polynomials

Question

I am having issues with calculating things like this: $3x^2 + 2 \mod x^2 + 3$ How would I got about this? I somehow couldn't find information anywhere.

Answer 1

Note that $a\equiv b\bmod c$ if $a-b$ is a multiple of $c$. Generally, the result of $a\bmod b$ is the unique value $x$ such that the degree of $x$ is less than the degree of $b$, and $a\equiv x\bmod b$.

To remove the quadratic term from $a$, we can subtract $3$ times the modulus. So, the answer is $$3x^2+2-3(x^2+3)=-7$$