Exercise 5.5.E Let $A = \sum_{n\in\mathbb{Z}} A_n$ be a graded commutative ring with a $\mathbb{Z}$ type grading. Let $f \in A_d, d > 0$. Suppose $f$ is invertible. Let h-$Spec(A)$ be the set of homogeneous prime ideals of $A$. Let $\psi\colon$ h-$Spec(A) \rightarrow Spec(A_0)$ be the map defined by $\psi(P) = P \cap A_0$. Then $\psi$ is a bijection.
His hint is as follows. Let $P_0 \in Spec(A_0)$. Define $Q_n = \{x \in A_n| x^d/f^n \in P_0\}$. Let $P = \sum_{n\in \mathbb{Z}} Q_n$. Show that $x \in Q_n$ if and only if $x^2 \in Q_{2n}$. Show that if $x, y \in Q_n$, then $x^2 + xy + y^2 \in Q_{2n}$ and hence $x + y \in Q_n$. Then show that $P$ is a homogeneous ideal. And then show that $P$ is prime.
I don't understand why $x^2 + xy + y^2 \in Q_{2n}$ if $x, y \in Q_n$ and hence $x + y \in Q_n$.