Prove without using complex numbers

Question

Prove without using complex numbers that $f:\mathbb{R}→\mathbb{R}$ does not exist such that $f(f(x))=x^2+4x+3$. Note that for example if $g(g(x))=x^2-2$ then $g(g(x))$ has two real fixed points -1 and 2; also $g(g(g(g(x))))$ has 4 fixed points -1 and 2 and $\frac{\sqrt5-1}{2}$ and $-\frac{\sqrt5+1}{2}$. Now we can see with some accuracy that there is no such function $g$ that $g(g(x))=x^2-2$. But $f(f(x))=x^2+4x+3$ and $f(f(f(f(x))))$ (and similar functions) does not have real fixed points. If these problems are solved, the questions like the following link can be resolved decisively: On the functional square root of $x^2+1$ I have a solution using complex numbers but I tried to find a preliminary solution to it but I did not succeed. Thanks