RO-NMO 2005 Problem 82

For a positive integer $n$ , written in decimal base, we denote by $p(n)$ the product of its digits.

a) Prove that $p(n) \leq n$ ; b) Find all positive integers $n$ such that $$10p(n) = n^2+ 4n - 2005.$$

Eugen Păltănea