Is $\frac{x^2 + 2x}{x}$ a polynomial? 
Is $$\dfrac{x^2 + 2x}{x}$$ a polynomial?  

 A: Some people would say that the rational number $7/1$ is not really equal to the integer $7$, but merely canonically identified with it.  But (after reaching a certain level of sophistication) mathematicians say that $7/1$ and $7$ are indeed equal.  

Some people would say that the rational function
$$
\frac{x^2+2x}{x}
\tag{*}$$
is not really equal to the polynomial
$$
x+2, 
\tag{**}$$
but merely canonically identified with it.  But (after reaching a certain level of sophistication) mathematicians say that (*) and (**) are indeed equal. 
A: Well, it's a polynomial in the variable $t=\tfrac{x^2+2x}{x}$... But you probably mean a polynomial in the (real?) variable $x$. What precise definition of polynomial are you using?

I would say no, because it is not of the form
$$a_0+a_1x+a_2x^2+\ldots+a_nx^n$$
for any $n \in \mathbb{N}$ and real numbers $a_i$ ($0 \le i \le n$).
Note that you cannot just simplify
$$\frac{x^2+2x}{x} = x+2$$
as this equality is only valid for non-zero $x$, so not for all $x$.
