<s>[INST] Define $P(x)\in [\mathbb{R}]$ is a polynomial that satisfy $P(x).P(y)=P(\frac{x+y}{2})^{2}-P(\frac{x-y}{2})^{2}$ Define $P(x)\in [\mathbb{R}]$ is a polynomial that satisfy
$P(x).P(y)=P(\frac{x+y}{2})^{2}-P(\frac{x-y}{2})^{2}$     (1)
Some notes may help you.
It is very clear that with y=0 then (1)$\Leftrightarrow P(x).P(0)=0$
We just need to consider  $P(x)\neq 0$ so $P(0)=0$.
With $y=-x$ then (1)$\Leftrightarrow P(x)=-P(-x)$
 [/INST] If $P$ is constant, then it is equal to $0$:

Otherwise:
When $X \to \infty$
$$
\begin{align}
P(X)^2 = a_n^2 X^{2n} + a_na_{n-1} X^{2n-1} + o(X^{2n-1})
\end{align}
$$
Let $y\in\Bbb R$, with $P(y)\neq 0$. When $x\to +\infty$
$$
\begin{align}
P(x+y)^2 - P(x-y)^2 &= a_n^2 (x+y)^{2n} + a_na_{n-1} (x+y)^{2n-1} \\
&- a_n^2(x-y)^{2n} - a_na_{n-1}(x-y)^{2n-1} + o(x^{2n-1})\\
& =a_n^2x^{2n} + a_n^2 (2n)x^{2n-1}y  + a_na_{n-1} x^{2n-1} \\
&-a_n^2x^n + a_n^2 (2n)x^{n-1}y - a_na_{n-1}x^{2n-1} + o(x^{2n-1})\\
&=4a_n^2 ny x^{2n-1} + o(x^{2n-1})\\
&\approx 4a_n^2 ny x^{2n-1}
\end{align}
$$
And 
$$
\begin{align}
P(2x) P(2y) &\approx P(2y)a_n 2^nx^n \\
4a_n ny x^{2n-1} &\approx P(2y)2^nx^n
\end{align}
$$
Meaning that $2n-1 = n$, so $n=1$ and $4a_1y = P(2y)2$
So $P(2y) = 2a_1y$
$$P(x) = a_1 x $$
Where $a_1$ can be any non zero constant.
</s>