There are various methods to do this problem, the common two are substitution or elimination. Substitution is where you will set one equal to either x or y then put that into the second equation in place of x or y. Elimination gets rid of x or y by adding or subtracting both equations. I will show you elimination.
Call "x+4y=-9" 1'
Call "-2x-5y=12" 2'
3' is 2*1' (thats 2 times equation 1) --> 2x+8y=-18
then add 3' and 2' together, this eliminates x.
3y=-6 so y=-3
then put y into equation 1'
x+ (4*-3) = -9
x - 12 = -9
x = 3
Answer x=3 y=-3