What Is Quadratic (2nd Order) Taylor Approximation?

Find the sum of all positive integers $1\leq k\leq 99$ such that there exist positive integers $a$ and $b$ with the property that $$x^{100}-ax^k+b=(x^2-2x+1)P(x)$$ for some polynomial $P$ with integer coefficients.

(Source: NIMO 18)