Let $A,B\in M_n (\mathbb R), \lambda\in \sigma(B), \alpha \in \sigma (B)$ and $<x,y>$ be inner product show $<Ax+By,x>=\lambda||x||^2+\alpha<x,y>$

Question

Let $A,B\in M_n (\mathbb R), \lambda\in \sigma (A), \alpha \in \sigma (B)$ and $<x,y>$ be inner product show $<Ax+By,x>=\lambda||x||^2+\alpha<x,y>$ where $x,y\in \mathbb R^n$ are eigenvectors associated with $\lambda,\alpha$ respectively

I'm really confused I'm not sure what this problem means when they say $\lambda\in \sigma(B), \alpha \in (B)$ - what is sigma B?

In this problem here the solution checked the three conditions and I thought that was I was supposed to do?