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Suppose given polynomials e,q,h,r in R[x], p in R (R a ring), how can I use Sage to find f in R[x] so fe = qh + r (mod p)?

Similarly, given f,g in R[x] (gcd(f,g)=1), what function can I use to compute s,t in R[x] so sf + gh = 1 (mod p) ?

Suppose given polynomials e,q,h,r $e,q,h,r$ in R[x], p in R $R[x]$, $p \in R$ (R a ring), how can I use Sage to find f $f$ in R[x] $R[x]$ so f$fe = qh + r (mod p)?(\text{mod} p)$?

Similarly, given f,g $f,g$ in R[x] (gcd(f,g)=1), $R[x]$ with $\text{gcd}(f,g)=1$, what function can I use to compute s,t $s,t$ in R[x] $R[x]$ so s$sf + gh = 1 (mod p) (\text{mod} p)$ ?

Suppose given polynomials $e,q,h,r$ in $R[x]$, $p \in R$ (R a ring), how can I use Sage to find $f$ in $R[x]$ so $f$f e = qq h + r (\text{mod} (\text{mod } p)$?

Similarly, given $f,g$ in $R[x]$ with $\text{gcd}(f,g)=1$, what function can I use to compute $s,t$ in $R[x]$ so $s$s f + gg h = 1 (\text{mod} (\text{mod } p)$ ?