use the given equation itself substitute x=1 and y=-2 i.e.
a(1)^3+b(1)^2-(1)+5=(-2) which gives the second equation a+b=-6
Then simultaneous equations your way.

use the given equation itself substitute x=1 and y=-2 i.e.
a(1)^3+b(1)^2-(1)+5=(-2) which gives the second equation a+b=-6
Then simultaneous equations your way.