Consider the rational polynomials $h,f$:
$h:=x^8+x^6-3x^4-3x^3+8x^2+3x-5$
$f:=3x^6+5x^4-4x^2-9x+2$
Compute polynomials $r,t \in \mathbb{Q}[x]$ such that
$r\equiv tf \mod{h}$ with $\deg r < 4, \deg t \leq 4$.
The hint is to write down the extended remainder sequence produced when executed the extended Euclidean algorithm on the input $(h,f)$ and to decide which of the triplets $(r_i,s_i,t_i)$ determine rational function approximations, i.e., a pair $(r,t)$ of polynomials in $\mathbb{Q}[x]$ such that $\deg r < n, \deg t \leq n - \deg r$ and so that
$\gcd(t,h)=1$ and $rt^{-1} \equiv f \mod{h}$
I an compute the extended Euclidean algorithm using computer algebra however I'm not sure what's meant by the "extended remainder sequence" or how to continue with this result.