If $d$ divides $a^4+a^3+2a^2-4a+3$, prove that $d$ is a fourth power modulo $13$

If $d$ divides $f(a)=a^4+a^3+2a^2-4a+3$, prove that $d$ is a fourth power modulo $13$.

$f(a)\equiv{(a-3)}^4\pmod 13$. But how can we prove any divisor of $f(a)$ is a fourth power? If we prove that any prime divisor $p$ of $f(a)$ is a fourth power modulo $13$, we would be done as the fourth powers form a group under multiplication modulo $13$.

If we can write $f(a)=P(a)^2-13Q(a)^2$ for some polynomials $P(a)$ and $Q(a)$, $p$ divides $f(a)$ implies $13$ is a square modulo $p$ and quadratic reciprocity will imply $p$ is a square modulo $13$. I could not find suitable $P(a)$ and $Q(a)$ but I think this will help. To prove fourth power, I think quartic reciprocity will help, but I don't know.

Any help will be appreciated.

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