I don't have a cool name for this

I have a proper solution. The roots are integers. We know that x+y=-p and xy=444p. So xy is a multiple of p. But p divides xy if p divides either x or y. Let's say that p divides x. Which means that x=np, but we know that x+y=-p which gives that y=-p-np=-(n+1)p. Now we'll use the product: xy=444p. Which means that -n(n+1)p^2=444p. Which means that p^2 divides 444p, which gives p divides 444. Since p is prime and a divisor of 444 then p can only be 2, 3 or 37. Then plug those values of p in the equation and check if the roots are integers. The discriminant is a square number only when p=37. The other solutions are incomplete or simple guesses.

We can rewrite the equation in two other forms, as : $x^{2}+px+(-148) \times3p=0$ or $x^{2}+px+148 \times(-3p)=0$ . It is clear that $-148+3p=p$ and $148+(-3p)=p$ , then we get $p=-74$ and $p=37$ respectively. Since $p$ is prime, hence $\boxed{p=37}$ .

Simple middle term splitting method can help to solve it x^2 +px-444p=0 444=37×4×3×p 37×4-3p=p p=37

I have no proper solution...Discriminant id p^2+1776p...Use a calculator and keep substituting primes to make it a perfect square....personally...i dont think the question to be "cool"...is there a better way??