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...)

Hardy is fairly robust on this issue (and while possibly slightly at variance with current set-theoretic dogma, nevertheless quite consistent with the standard calculus interpretations):

The function $\frac{x^2-1}{x-1}$ has no value for $x=1$; for $x=1$, $\frac{x^2-1}{x-1}$ is strictly and absolutely meaningless. The fact that its limit for $x=1$ is $2$ is entirely irrelevant. The functions $\frac{x^2-1}{x-1}$ and $x+1$ are different functions. They are equal when $x$ is not equal to $1$. Similarly the function $y=\frac{x}{x}$ is $=1$ when $x \neq 0$ and undefined for $x=0$. To calculate $f(x)$ for $x=0$ we must put $x=0$ in the expression of $f(x)$ and perform the arithmetical operations which the form of the function prescribes, and this we cannot do in this case.

(Mathematical Gazette 1907, 4 pp. 13–14)
