Question

18x -59xy-220y

Answer 1

18x²-59xy-220y²=(2x-11y)(9x+20y). Is that what you meant?

How to work it out:

factor pairs (x-set) for 18=(1,18), (2,9), (3,6); (a,c) values (see below).

factor pairs (y-set) for 220=(1,220), (2,110), (4, 55), (5,44), (10,22), (11,20); (b,d) values.

(ax-by)(cx+dy)=acx²+xy(ad-bc)-bdy². Note the opposite signs in the factors.

The table below contains all possible combinations of the coefficients a, b, c, d. The key column is ad-bc. This is the coefficient of the xy term. We want the number -59.

The highlighted row contains -59 because ad-bc=-59. So since a=2, b=11, c=9, d=20 the expression factors: (2x-11y)(9x+20y).

a	b	c	d	ad	bc	ad-bc
1	1	18	220	220	18	202
2	1	9	220	440	9	431
3	1	6	220	660	6	654
18	1	1	220	3960	1	3959
9	1	2	220	1980	2	1978
6	1	3	220	1320	3	1317
1	2	18	110	110	36	74
2	2	9	110	220	18	202
3	2	6	110	330	12	318
18	2	1	110	1980	2	1978
9	2	2	110	990	4	986
6	2	3	110	660	6	654
1	4	18	55	55	72	-17
2	4	9	55	110	36	74
3	4	6	55	165	24	141
18	4	1	55	990	4	986
9	4	2	55	495	8	487
6	4	3	55	330	12	318
1	5	18	44	44	90	-46
2	5	9	44	88	45	43
3	5	6	44	132	30	102
18	5	1	44	792	5	787
9	5	2	44	396	10	386
6	5	3	44	264	15	249
1	10	18	22	22	180	-158
2	10	9	22	44	90	-46
3	10	6	22	66	60	6
18	10	1	22	396	10	386
9	10	2	22	198	20	178
6	10	3	22	132	30	102
1	11	18	20	20	198	-178
2	11	9	20	40	99	-59
3	11	6	20	60	66	-6
18	11	1	20	360	11	349
9	11	2	20	180	22	158
6	11	3	20	120	33	87

In practice, you probably don't need the full table, as many possibilities can be mentally perceived as unlikely. Nevertheless, this method does ensure that rational factors can always be found if they exist.