Is this polynomial familiar? - II

See that one obvious solution you can see is $f(x)=x^2$ for $x \in [1,10]$

Because we have apparently $f(x)=x^2$ given only for the specific range of $x$ , we can set our $f(x)$ as $f(x)=x^2 +(x-1)(x-2)(x-3)...(x-9)(x-10) g(x)$

This will follow our needed thing, that $f(x)=x^2$ for $x \in [1,10]$ . So some bracket will get $0$ , making $f(x)=x^2$

See that here $g(x)$ be any any polynomial, the condition holds ! And $g(x)$ has infinite possibilities, so answer is as in the note , $\boxed{0}$

Polynomials of form $P(x) = x^2 + Q(x)(x-1)(x-2)\cdots(x-10)$ satisfy the conditions. (Actually they are the only polynomials to satisfy the conditions.)

i tried my luck and BINGO !!