Why does this Polynomial have two valid factored forms?

Question

When factoring $3 + 4a - 7a^2$ why are there two possible factors? $(-7a -3)(a - 1)$ and $(3 + 7a)(1 - a)$. What is the principle? I assume that the two factors are inverse in some way.

Answer 1

(-7a -3) and (3 + 7a) are "equivalent" in the sense that they differ by a multiplicative constant (which is -1 in this case). The same is true for (a - 1) and (1 - a).

Other instance is: Ax + By + C = 0 and - Ax - By - C = 0 are two different equations but they represent the same straight line.

Answer 2

The only difference between the two is one is multiplied by -1. If you were to find the solution for a. You get a= -3/7 for both.

Ex 1.

-7a-3 = 0

a = -3/7

Ex 2.

3+7a=0
a = -3/7