To complement Dima's answer, you can use // and %
to get the quotient and the remainder individually.

sage: _.<z0> = GF(2)[]

sage: K.<z> = GF(2^8, modulus=z0^8+z0^4+z0^3+z0+1)

sage: R.<x> = K[]

sage: l = x^4 + 1

sage: c = (z+1)*x^3+x^2+x+(z)

sage: l.quo_rem(c)

((z^7 + z^6 + z^5 + z^4 + z^2 + z)*x + z^6 + z^4 + z,

 (z^7 + z^5 + z^2)*x^2 + (z^7 + z^5 + z^2 + 1)*x + z^7 + z^5 + z^2 + 1)

sage: l // c

(z^7 + z^6 + z^5 + z^4 + z^2 + z)*x + z^6 + z^4 + z

sage: l % c

(z^7 + z^5 + z^2)*x^2 + (z^7 + z^5 + z^2 + 1)*x + z^7 + z^5 + z^2 + 1

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