Do i need to always take out factor of -1 when solving a quadratic in form $ax^2 + bx + c$ when $a$ is negative?

Question

In my textbook I am advised to always take out the factor of -1 if the quadratic is in the form of $ax^2 + bx + c$ when $a$ is negative.

For example:

$-4x^2 + 4x + 3$

My question is, is this compulsory or could it be resolved without taking out factor -1 at the beginning, as the results are the same?

like this:

$-4x^2 + 4x + 3 = -4x^2 + 6x - 2x + 3 = 2x(-2x + 3) + (-2x +3) = (2x + 1)(-2x + 3)$

and then i can take out $-1$

$-(2x + 1)(2x - 3)$

Is this a correct way to resolve it or should I take out the factor of -1 at the begining and then resolve the equation,

$-4x^2 + 4x + 3 = -(4x^2 - 4x - 3) = -(2x + 1)(2x - 3)$

Answer 1

Yes, you are absolutely correct and your procedure is perfect. And there is no such rule in algebra which states that this cannot be done.

Answer 2

As I already commented, it is quite the opposite. Not only there is no rule in algebra that prohibits this, there is an explicit rule that this is allowed and will end in the same result!

If you are interested, you might want to check rings and associative algebras for precise list of rules that are allowed in manipulations when dealing with quadratics.

Your textbook advises factoring out $-1$ first because it is more convenient and probably easier to notice correct factorization. Many mistakes in algebraic manipulations are due to mistreating the minus sign, as we all know, so if you get it out of the way in the beginning, there is (arguably) less chance that mistake will happen. Far from it that this is required.