What polynomial could you subtract from 5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4 to give a difference of 19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4?
-6-2 = -8
5-(-3) = 8
...
so the subtrahend starts out with
2x^4-3x^3...
Just look at the coefficients of like powers. They threw you a curve by scrambling up the order of the powers. The first step would be to write the polynomials in descending order, so it's easier to line up the coefficients.