Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$

Let $a_1,a_2,\cdots$ be a strictly increasing sequence on positive integers. Is it always possible to partition the set of natural numbers $\mathbb{N}$ into infinitely many subsets with infinite cardinality $A_1,A_2,\cdots$, so that for every subset $A_i$, if we denote $b_1<b_2<\cdots$ be the elements of $A_i$, then for every $k\in \mathbb{N}$ and for every $1\le i\le a_k$, it satisfies $b_{i+1}-b_{i}\le k$?

Let $G$ be the centroid of a triangle $\triangle ABC$ and let $AG, BG, CG$ meet its circumcircle at $P, Q, R$ respectively. Let $AD, BE, CF$ be the altitudes of the triangle. Prove that the radical center of circles $(DQR),(EPR),(FPQ)$ lies on Euler Line of $\triangle ABC$. Proposed by Ivan Chai, Malaysia.