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p=7 R=PolynomialRing(IntegerModRing(7),'x') x=R.gen()

I'm not sure if f exist

Maybe f=(q*h+r)//e

Or you want something like this? q=f//g r=f.mod(g)

As for the next question:

f=x^14 + 4*x^7 + 7

g=x^3+1

xgcd(f,g)