Pair of straight lines problem

Question

Find the lines whose whose combined equation is $6x^2+5xy-4y^2+7x+13y-3=0$ using the concept of parallel lines through the origin.

Now, I understand that the homogeneous part of the given equation represents another pair of lines passing through the origin. Thus:

$6x^2 + 5xy - 4y^2 = 0 \implies (3x+4y)(2x-y)=0$

Now, does this result help? What do I do next and why?

Answer 1

Now you've got the part of degree 2, but you should also capture the lower degree part, so you can use the ansatz $$6x^2+5xy-4y^2+7x+13y-3 = (3x+4y+b)(2x-y+a)$$ expand it, and solve for $a$ and $b$ by comparing coefficients.