What does it mean when you get a constant remainder when finding the gcd of 2 polynomials in a finite field mod n?

Question

Use the Euclidean Algorithm to find the GCD of $3x^2+1$ and $x+1$ in $\mathbb{Z}_5[x]$.

What I got was this:

$3x^2+1 = (x+1)(3x-3) + 4$

$x+1 = (4)(4x+4) + 0$

Since 4 was the last non-zero remainder, that is our GCD for the two polynomials. But, what does this mean?