I've recently been delving into summation formulas involving quadratic residues modulo prime numbers and am seeking strategies or insights for a formal proof. Thus far, my efforts have focused on computational verification. However, I am somewhat at a loss on how to transition from these observations to a rigorous proof. For example, I've checked that for all primes $43 \leq p < 10^5$ such that $p \equiv 3 \ (\text{mod} \ 8 )$, the following holds:
$$\sum\limits_{\substack{1 \leq n \leq p-1 \\ (\frac{n}{p})=1}} {\frac{(1+12n+n^2)^3}{n(1-n)^4}} \equiv -16 \ (\text{mod} \ p)$$
Here, the summation is over all quadratic residues $1 \leq n \leq p-1$.
I am specifically looking for strategies or theorems that could be applicable in proving this kind of summation formula or examples of similar proofs.
Any help or pointers towards relevant literature or theorems would be greatly appreciated.