3x^2 + 10xy + 8y^2 + 14x + 22y + 15 = 0
3x^2 is factorable only as 3x and x.
15 can be 15 * 1 or 3 * 5.
As a first guess, I place 5 and 3 in that order.
The coefficients of y are unknown so far.
= (3x + Ay + 5)(x + By + 3)
= 3x^2 + 3Bxy + 9x + Axy + ABy^2 + 3Ay + 5x + 5By + 15
(3B + A)xy = 10xy
(3A + 5B)y = 22y
A + 3B = 10
3A + 5B = 22
-3A - 9B = -30
3A + 5B = 22
4B = 8
B = 2
A + 3B = 10
A + 6 = 10
A = 4
3x^2 + 10xy + 8y^2 + 14x + 22y + 15 = (3x + 4y + 5)(x + 2y + 3)
The product of (3x + 4y + 5)(x + 2y + 3) is indeed
3x^2 + 10xy + 8y^2 + 14x + 22y + 15
The lines are
3x + 4y + 5 = 0
x + 2y + 3 = 0
