The polynomial $x^4+2x^3-3x^2$ + ax + b leaves the remainder 5 on division by $(x-1)$ and the remainder 9 on division by $(x+1)$. The remainder when it is divided by $(x-2)$ is:

The remainder theorem gives p(1) = 1 + 2 − 3 + a + b = a + b = 5 and p(−1) = 1 − 2 − 3 − a + b = b − a − 4 = 9, so b − a = 13. Solving, a = −4 and b = 9. Then p(2) = 16 + 16 − 12 + 2(−4) + 9 = 21. (37 uses a = +4, a sign slip when solving; 34 swaps the values of a and b; 5 just re-quotes the first given remainder.) Check p(1) = 1 + 2 − 3 − 4 + 9 = 5 and p(−1) = 1 − 2 − 3 + 4 + 9 = 9.