Can a quotient of polynomials be simplified before been analyzed? Having the function $f(x) =\frac{X^3+3X^2}{2X^2+4x}$, why it is not the same to analyze $\frac{X^2+3X}{2X+4}$, if it verifies $\frac{X^3+3X^2}{2X^2+4x}=\frac{X^2+3X}{2X+4}$ ? In this case, the first one has only one root, while the second one has another one in $0$.
In general, can a function be simplified before been analyzed? (I mean, find roots, continuity, maxima and minima...)

The function under analysis can generally be simplified. But note that when you simplify, canceling top and bottom of the same fraction is only valid if you are not dividing by zero.
