Prove that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p

Question

I am trying to prove that the congruence $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p. I proved it using primitive root, but my professor in number theory told me that it can be more directly done using the hasse-weil theorem in the theory of elliptic curves, but i cant do it. Would someone kindly show me how to make use of the hasse-weil theorem, please? Thank you in advance.