Did I misinterpret the sign attached to a term when factoring?

[find_the_fun]

I wanted to factor \(\displaystyle xy+x-2y-2\)
I got \(\displaystyle x(y+1)-2(y-1)\) and got stuck

I tried somethings out and noticed \(\displaystyle (x-2)(y+1)=xy+x-2y-2\) so how come I did get stuck? Did I extract the \(\displaystyle 2\) incorrectly with the sign? For example should I have interpreted it at \(\displaystyle -2y-2\) not \(\displaystyle 2y-2\)?

 

[MarkFL]

I wanted to factor \(\displaystyle xy+x-2y-2\)
I got \(\displaystyle x(y+1)-2(y-1)\) and got stuck

I tried somethings out and noticed \(\displaystyle (x-2)(y+1)=xy+x-2y-2\) so how come I did get stuck? Did I extract the \(\displaystyle 2\) incorrectly with the sign? For example should I have interpreted it at \(\displaystyle -2y-2\) not \(\displaystyle 2y-2\)?

Yes, you simply factored incorrectly.

\(\displaystyle xy+x-2y-2=x(y+1)-2(y+1)=(x-2)(y+1)\)

When you factor a negative factor from a group of terms, the signs of everything within the resulting parentheses must change.

 

[find_the_fun]

Yes, you simply factored incorrectly.

\(\displaystyle xy+x-2y-2=x(y+1)-2(y+1)=(x-2)(y+1)\)

When you factor a negative factor from a group of terms, the signs of everything within the resulting parentheses must change.

But \(\displaystyle 2y-2\) can be factored to \(\displaystyle 2(y-1)\). Since \(\displaystyle 2y-2\) appears in the equation can't we say \(\displaystyle ...-(2y-2)=...-(2(y-1))\)?

 

[MarkFL]

It is $-2y-2$ that appears in the expression...and both terms have $-2$ as a factor. :D

 

[I like Serena]

But \(\displaystyle 2y-2\) can be factored to \(\displaystyle 2(y-1)\). Since \(\displaystyle 2y-2\) appears in the equation can't we say \(\displaystyle ...-(2y-2)=...-(2(y-1))\)?

That's the point. \(\displaystyle 2y-2\) does not appear in the equation.
You can read the equation as:
$$xy+x−2y−2 = xy+x+(−2)y+(−2)$$

As you can see $(−2)y+(−2)$ is definitely different from $2y-2$.

From here we can get to:
$$xy+x+(−2)y+(−2) = x(y+1)+(−2)(y+1)$$

Which can also be written as:
$$x(y+1)−2(y+1)$$

 

[find_the_fun]

I guess I see what I did wrong, I did something like, took a -2 inserted a bracket and pretend it was positive. I try to understand exactly what went wrong in hopes of not making the same mistake again.