How do we know what sign goes outside of the bracket when factoring binomials?

I'm assuming you're asking about negative or positive signs here. When I factor binomials, it depends on what would make factoring simpler for me. And how you're factoring the binomial too.

More often than not, it's going to be positive. For example, 4x^2 - 2x = 0. That would factor out to be 2x(2x-1) = 0. The sign outside is going to be positive because it's simpler to solve that way. Here, x is equal to 0 and equal to 1/2. Taking out a negative in this case would complicate the solving process.

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