Is $P$ unique?

Relevant wiki: Diophantine Equations - Solve by Factoring

Let $-a$ , $-b$ and $-c$ be the roots of $P$ where $a$ , $b$ and $c$ are positive integers.

Then $P(x)$ can be written in the form $A(x+a)(x+b)(x+c)$

Now $P(1)=A(1+a)(1+b)(1+c)=3553=11\times17\times19$

But as $a$ , $b$ and $c$ are positive integers then $1+a$ , $1+b$ and $1+c$ are all positive integers greater than 1.

Therefore $(1+a,1+b,1+c)=(11,17,19)$ (in some permutation) and $A=1$

Then $(a,b,c)=(10,16,18)$ (in some permutation)

Thus $P(x)=(x+16)(x+10)(x+18)$ and $P(-1)=(-1+16)(-1+10)(-1+18)=(15)(9)(17)=2295$