I'm working on this question...
What is the multiplicative inverse of $3+2[x]$ in the field $\mathbb{Q[x]}/(x^2+7)$
I have been searching throughout this website on how to do such problems related to this one because there aren't any examples of it on my book. According to what I have found on similar questions, I would have to, in some way, implement Euclidian division to $x^2+7$. Since $x^2+7$ is irreducible, the $gcd$ with any other polynomial will yield $gcd = 1$. What would I do next, if I'm even in the correct path?
You can do it directly. You want to find a polynomial $p(x)$ such that $p(x)(3+2x)=1$ in your field. Because we are taking out $x^2+7$ we might guess that a linear polynomial will work because we can get rid of quadratic terms. So we write $(ax+b)(3+2x)=2ax^2+(3a+2b)x+3b=(3a+2b)x+(3b-14a)=1$
This gives us two simultaneous equations $3a+2b=0, 3b-14a=1$ which we can solve for $a$ and $b$. I essentially used the Euclidean algorithm when I replaced $x^2$ by $-7$.