Let $A,Bin M_n (mathbb R), lambdain sigma(B), alpha in sigma (B)$ and $$ be inner product show $=lambda||x||^2+alpha$

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

I’m really confused I’m not sure what this problem means when they say $lambdain sigma(B), alpha in (B)$ – what is sigma B?

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