Q:

$\mathbb{R} [ X ] / ( X^ 2 + 2 )\cong \mathbb{C}$?

I am puzzled since in my eyes I cannot prove that there can be a function which is homomorphic regarding multiplication.
Do you know any functions that could be?
I was thinking that the quotient field has the format: 
< ax+b | x^2 = -2> 
Am I missing something?

A:

Hint: Consider the map from $\mathbb R[x]$ into $\mathbb C$ defined by $p(x)\mapsto p\left(\sqrt 2\,i\right)$.

