Using only elementary methods known to a middle schooler taking his first algebra course, is there a triplet of integers $(a, b, c)$ such that, for every prime number $p$, $(5a + 4)p^2 + (5b + 3)p + 2c$ is a prime number?
I know that if an even integer is a prime number, then it is $2$.
No such $a,b,c$ exist.
Let $f(x)=(5a+4)x^2+(5b+3)x+2c$. Let $f(3)=q$. Then $f(kq+3)$ is a multiple of $q$ for all $k$. And by Dirichlet's Theorem $kq+3$ is prime for infinitely many $k$.
Now it's not possible to prove Dirichlet's Theorem by middle school methods, but at least the statement of Dirichlet's Theorem is quite simple. It says that if $\gcd(a,b)=1$ then there are infinitely many $n$ such that $an+b$ is prime.