6. 144 Find all infinitely differentiable functions \( f: R \rightarrow R \) that satisfy
$$f(x+y) \equiv f(x)+f(y)+2 x y, x, y \in R$$
[Solution] In the identity given in the problem, let $x=y=0$ to get $f(0)=2 f(0)$, which means $f(0)=0$. For any $x \in \mathbb{R}$, from the identity
$$f(x+y)-f(x) \equiv f(y)+2 x y, \quad y \in \mathbb{R}$$

we get
$$\begin{aligned}
f^{\prime}(x) & =\lim _{y \rightarrow 0} \frac{f(x+y)-f(x)}{y} \equiv \lim _{y \rightarrow 0} \frac{f(y)+2 x y}{y} \\
& =2 x+\lim _{y \rightarrow 0} \frac{f(y)-f(0)}{y}=2 x+f^{\prime}(0)
\end{aligned}$$

Therefore, the function we seek satisfies
$$f(x)=f(x)-f(0)=\int_{0}^{x} f^{\prime}(x) d x=x^{2}+f^{\prime}(0) x$$

Note that, for each $a \in \mathbb{R}$, the function $f(x)=x^{2}+a x$ satisfies the identity given in the problem.