We look for rational answers first by noting the factors of 12 and 7.
Factors of 12 in pairs (1,12), (2,6), (3,4).
Factors of 7 are (1,7).
We are looking to factor in this form (aw±b)(cw∓d)=12w²+25w-7=0.
Since we know the factors of 7 we can put b=1 and d=7.
If we expand the parentheses we get:
12w²±w(7a-c)-7=0, and we want 7a-c=25 where ac=12. If we use (c,a)=(3,4), we get 28-3=25.
Therefore 12w²+25w-7=12w²+w(28-3)-7=(4w-1)(3w+7)=0, w=¼ or -⁷⁄₃.