Finding polynomials gcd

We need to find the inverse of 2x+5 in Q[x]/(x^2-2)

Now, I know what we need is to find the inverse of 2x+5, i.e., we need

(2x+5)u(x)+(x^2-2)(v(x))=1, and we need to find these u(x) and v(x).

I know you use the Euclidean Algorithm, and I can do it for integers, but this is confusing me.

Surely x^2-2=(2x+5)(1/2x-5/4)+(33/4) by long division, and I know we have to use back substitution now, but it confuses me with polynomials.

How exactly would I do this back substitution.

I know the answer is supposed to be -2/17 x+5/17, but I do not see how we get that .

Thank you.