Solution
Factor out 4x from 12x^{ 3} + 196x^{ 2}y + 392xy^{ 2}
4x(3x^{ 2} + 49xy + 98y^{ 2})
Front coefficient is 3. Factors are: 1, 3
Back coefficient is 98. Factors are: 1, 2, 7, 14, 49, 98
Now test combinations of these factors to get the middle value of 49xy.
For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to 49xy.
Test: (x + y)(3x + 98y) => 98xy + 3xy = 101xy
Test: (x + 2y)(3x + 49y) => 49xy + 6xy = 55xy
Test: (x + 7y)(3x + 14y) => 14xy + 21xy = 35xy
Test: (x + 14y)(3x + 7y) => 7xy + 42xy = 49xy This combination matches.
Result: 12x^{ 3} + 196x^{ 2}y + 392xy^{ 2}
