x^2 + 7xy + 12y^2 + 2x + 6y

collect the expression in terms of 'x' and look to factor the constant term with respect to 'y':

x^2 + x(7y+2) + (12y^2 + 8y)

factor '4y' out of '12y^2 + 8y':

x^2 + x(7y+2) + 4y(3y+2)

The factors 4y(3y+2) that sum to 7y+2 are x(3y+2) and 4xy:

x^2 + x(3y+2) + 4xy + 4y(3y+2)

factor pairs of terms by grouping:

x(x+4y) + (3y+2)(x+4y)

pull out a common factor, 'x+4y':

(x+4y)(x+3y+2)