gcd and lcm - GCD of $X^4+5X^3+3X^2+X$ and $X^2+1$ ( Euclidean Algorithm + Finding the inverse of the polynomial)

It is obvious the GCD of those two is 1.

The thing that makes me post this question is that why when using Euclidean Algorithm the GCD seems to be $\frac{5}{4}$ ?

I want furthermore to find the inverse of the polynomial and i know i can do that when the GCD = 1 by using Extended Euclidean Algorithm. Yet as i've said i have 5/4.

Answer

So what? The conclusion (assuming that the computations are correct) is that $\frac54$ is a GCD of your polynomials. This is the same thing as saying that they are relatively prime.