I would like to ask a question that might seem ridiculous but I'm really willing to comprehend the what and why rather than simply memorizing some rule or pattern that applies "the best". And I hope it will help other people who have similar question.
So, the issue is as follows: I'm always getting confused when expanding or simplifying an expression when it comes to the signs of its terms.
Let's take this expression for example:
$$4(x^2 + 3x - 5)$$
I get 3 possible scenarios in my mind:
SOLUTION#1: I have to multiply each term by factor of 4 WITHOUT looking at their signs. It goes like:
initial expression: $$4(x^2 + 3x - 5)$$ expanding: $$(4 * x^2) + (4 * 3x) - (4 * 5)$$ result: $$4x^2 + 12x - 20$$
I didn't take into account the plus and minus signs standing close to the numbers when multiplying them by 4. Rather, I left them there as it was in the initial expression as if this group in parentheses didn't know about distribution (if I can say so).
SOLUTION#2: I have to multiply each term by factor of 4 WITH their signs taken into account. It goes like (this approach might hurt your eyes and perhaps the common sense (if it wasn't hurt by the very question already) so I apologize in advance):
initial expression: $$4(x^2 + 3x - 5)$$ distributing to the first term: $$(4 * (+x^2)) = + 4x^2$$ distributing to the second term: $$(4 * (+3x)) = + 12x$$ distributing to the third term: $$(4 * (-5)) = - 20$$ so we get: $$4x^2 + 12x - 20$$
SOLUTION#3: Most of the times I used to solve it this way:
initial expression: $$4(x^2 + 3x - 5)$$ expanding: $$(4 * x^2) + (4 * 3x) - (4 * (-5))$$ then I would get: $$4x^2 + 12x - (-20)$$ And here's where I would get completely stuck as to if I should leave it as is OR make it like this: $$4x^2 + 12x + 20$$
I know that SOLUTION#3 is a result of the lack of knowledge and confusion between different rules.
SOLUTION#1 and SOLUTION#2 yield the correct answers though.
Could anyone please explain which of these solutions is correct and must be applied?
Or both of the first solutions are correct and might be used interchangeably?
Or none of them actually make sense?
UPD: I guess that my problem is that I can't differentiate when signs indicate some operation or they belong to some number and indicate that it's positive or negative. I'm used to view plus and minus signs as indicators of both operations and that number is positive or negative. And it results into confusion like in SOLUTION#3.
Solutions $1$ and $2$ would be the best path to take.
Solution #$1$ uses the distributive property by using all positive numbers and then using the operations in sequence, i.e. $4(x^2 + 3x - 5) = 4\cdot x^2 + 4 \cdot 3 x - \color{red}{({4 \cdot 5)}} = 4x^2 + 12x - 20.$
Solution #$2$ uses the distributive property by multiplying each of the numbers according to the signs, and any negative values become minus signs, i.e. $4(x^2 + 3x - 5) = +4\cdot (+1x^2) + +4 \cdot (+3x) + \color {blue}{({+4\cdot -5)}}= 4x^2 + 12x - 20.$
The problem with solution #$3$ is that there is an extra minus sign added before $5$, and while $(-4) \cdot (-5) = +20$, it produces the incorrect answer $4x^2 + 12x + 20$.