Yeah, i liked this question because at first, i wasn't sure that it's ok to go from:
(a+b)x+ab = 3x + c
=> (a+b)x =3x ; c =ab.
But then i realized that it must be correct, since both describe ONE specific line.
And there can only be one linear equation for this line.
This one is easier then you made it. Here's what I always teach my students:
whenever you've got this polynomial situation, there are always four elements to figure out. It'll always look like:
x^2 + bx + c = (x + d)(x + e).
And we always know that d + e = b and de = c
And of course we know how to factor the first to make the second if we have b and c, and we know how to mulitply the second to make the first if we've got d + e.
So the point is, we have enough informaiton as long as we have any two of b, c, d, and e to always get the other two. It's an absolute. Give me any two of those numbers, I'll give you the other two.
So in any problem, just look for two of them, and you'll get the other ones. You don't have to think about it - just do it.
Check out these example from the OG to back that up: