Shorey and Tijdeman, On the greatest prime factors of polynomials at integer points, Compositio Mathematica, tome 33, no 2 (1976), p. 187-195 http://www.numdam.org/item?id=CM_1976__33_2_187_0 note that if $f$ is a polynomial with integer coefficients and at least two distinct roots, and we write $P(m)$ for the greatest prime factor of $m$, then $$ P(f(n))\gg\log\log n $$ where the implied constant depends only on the polynomial $f$. They say the degree $3$ case follows from work of Keates. In fact, they prove something a little stronger, but not nearly strong enough to get the $p>3\log n$ that OP asks for.

The reference is M Keates, On the greatest prime factor of a polynomial, Proc Edinb Math Soc (2) 16 (1969) 301-303.

I expect there have been improvements on this result.

Shorey and Tijdeman, "On the greatest prime factors of polynomials at integer points", Compositio Mathematica, tome 33, no 2 (1976), p. 187-195, MR424681, Zbl 0338.10040 note that if $f$ is a polynomial with integer coefficients and at least two distinct roots, and we write $P(m)$ for the greatest prime factor of $m$, then $$ P(f(n))\gg\log\log n $$ where the implied constant depends only on the polynomial $f$. They say the degree $3$ case follows from work of Keates. In fact, they prove something a little stronger, but not nearly strong enough to get the $p>3\log n$ that OP asks for.

The reference is M Keates, On the greatest prime factor of a polynomial, Proc Edinb Math Soc (2) 16 (1969) 301-303, DOI:0.1017/S0013091500012967, MR257034, Zbl 0188.10201.

I expect there have been improvements on this result.