If I have $2x^2+6x+2=0$ and $x^2+3x+1=0$ and then I multiply both sides of $x^2+3x+1=0$ by 2, I get $2x^2+6x+2=0$, which is the same as the first equation. However, when I graph $2x^2+6x+2$ and $x^2+3x+1$, they aren't the same graphs. Why is that?
It is true that $2x^2+6x+2=0$ and $x^2+3x+1=0$ are same equations but when you are plotting you are plotting the functions $y(x) =2x^2+6x+2$ and $y(x)=x^2+3x+1$ which are different functions. One is a scaled version of the other.
To have same roots you need to have the curves intersect $y=0$ at the same points, which these two curves would do.