2016/17 AHSMC Part 1 Problem 9

The remainder must be a first degree polynomial $Ax + B$ and if $Q(x)$ is the quotient then $$x^4 + x^5 + x^{10} + x^{17} + x^{100} = Q(x)(x^2-1) + Ax + B$$ for any real $x$. Taking $x = \pm 1$ in the above equation we obtain $A+B = 5$ and $-A+B = 1$ hence $A = 2$, $B = 3$ and thus the remainder is $2x+3$. The answer is (C).