Whenever you do something like:
$y^2 = 4y $ so $y^2/y = 4y/y $
Or $y (y+1)=2 (y+1) $ so $y (y+1)/(y+1)=2 (y+1)/(y+1) $
Ever SINGLE time, you absolutely MUST, with no exceptions EVER, you must acknowledge, the possibility that what you are dividing by might be zero and thus you can not do it, and you must make a case for this.
You must do this EVERY time.
So for example:
If you are solving $y^2 = 7y $ and you say:
$y^2 = 7y$
Then you have done it WRONG!!!
That is the wrong answer.
You must do it this way instead.
"$y^2 = 7y $"
"There are two cases we must consider.
Case 1: $y \ne 0$
And case 2: $y=0$.
If $y\ne 0$ then we can divide both sides by $y $ and
$y^2/y = 7y/y $
$y = 7$
BUT we must also consider case 2.
If $y= 0$ we can NOT divide by $y $. In this case $y=0$ so we are done.
Our answer is two possibilities: either $y=7$ or $y=0$.
To do $4y = y $ we MUST do two cases:
Case 1: $y \ne 0$. IF so (and this is an "if"; it might not be true) we can divide by $y $ and get $4=1$. As this is impossible, this case is not true. It is NOT the case that $y\ne 0$.
Case 2: $y = 0$.
This must be true.
So our answer is $y=0$ and we can not EVER divide by 0.
You must do these two cases every single £¥₩@ing time. If you don't you are doing it wrong. Period.