Let $P(n)$ be an irreducible polynomial of degree $2$ over the positive integers. Do there exist infinitely many positive integers $n$ such that $P(n)$ divides $n!$?


Edit: motivation by examples:
A) $p(n)=n^2+1$ (true, $21^2+1$ divides $21!$).  

B) $p(n)=n^2+n+1$ (true, $74^2+74+1$ divides $74!$).



Polynomial whose values divide $n!$
Here is a completely elementary proof, inspired by Pasten's comments.


Let $P(n)=an^2+bn+c$.


Take $n=a^5x^4+2a^3(ab+2a+1)x^3+a(2a^3c+a^2b^2+6a^2b+3ab+6a^2+5a+1)x^2+(ab+2a+1)(2a^2c+2ab+b+2a+1)x+a^3c^2+2a^2bc+abc+2a^2c+ac+c+ab^2+b^2+2ab+2b+a+1$


Then


$P(n)=P\_1(x)P\_2(x)P\_3(x)$


where


$$\begin{align\*}
P\_1(x)=&a^2x^2+abx+2ax+ac+b+1\\
\\
P\_2(x)=&a^4x^2+a^3bx+2a^3x+2a^2x+a^3c+a^2b+ab+a^2+2a+1\\
\\
P\_3(x)=&a^5x^4\\
&+2a^4bx^3+4a^4x^3+2a^3x^3\\
&+2a^4cx^2+a^3b^2x^2+6a^3bx^2+3a^2bx^2+6a^3x^2+4a^2x^2+ax^2\\
&+2a^3bcx+4a^3cx+2a^2cx+2a^2b^2x+ab^2x+6a^2bx+4abx+bx+4a^2x+2ax\\
&+a^3c^2+2a^2bc+abc+2a^2c+c+ab^2+b^2+2ab+b+a\\
=&n-(a^2x^2+abx+2ax+x+ac+b+1)
\end{align\*}$$


Clearly for $x$ large enough $P\_1(x), P\_2(x), P\_3(x)$ are distinct and less than $n$ in absolute value.
Since the product of 3 distinct numbers $\le n$ divides $n!$, the result follows.$\qquad\qquad\blacksquare$


The idea behind this proof is that always $P(x)\;|\;P(P(x)+x)$ and applying this idea twice allows to factor $P(\text{some polynomial})$ into 3 factors all small enough.


Here is the relevant Maxima code:



```
(%i1)   a*x*x+b*x+c;
(%o1)   ...
(%i2)   a*x^2+(b+1)*x+c;
(%o2)   ...
(%i3)   subst(a^2*x^2+a*b*x+2*a*x+x+a*c+b+1, x, (%o2));
(%o3)   ...
(%i4)   expand((%o3));
(%o4)   ...
(%i5)   subst(%o4, x, (%o1));
(%o5)   ...
(%i6)   expand((%o5));
(%o6)   ...
(%i7)   factor((%o6));
(%o7)   (a^2*x^2+a*b*x+2*a*x+a*c+b+1)*(a^4*x^2+a^3*b*x+2*a^3*x+2*a^2*x+a^3*c+a^2*b+a*b+a^2+2*a+1)*(a^5*x^4+2*a^4*b*x^3+4*a^4*x^3+2*a^3*x^3+2*a^4*c*x^2+a^3*b^2*x^2+6*a^3*b*x^2+3*a^2*b*x^2+6*a^3*x^2+4*a^2*x^2+a*x^2+2*a^3*b*c*x+4*a^3*c*x+2*a^2*c*x+2*a^2*b^2*x+a*b^2*x+6*a^2*b*x+4*a*b*x+b*x+4*a^2*x
+2*a*x+a^3*c^2+2*a^2*b*c+a*b*c+2*a^2*c+c+a*b^2+b^2+2*a*b+b+a)

```

