Thank you for reporting, we will resolve it shortly
Q.
Let $\lambda \in R$ and let the equation $E$ be $|x|^2-2|x|+|\lambda-3|=0$. Then the largest element in the set $S=$ $\{x+\lambda: x$ is an integer solution of $E\}$ is
$ | x |^2-2| x |+|\lambda-3|=0$
$ | x |^2-2| x |+|\lambda-3|-1=0$
$ (| x |-1)^2+|\lambda-3|=1$
At $\lambda=3, x =0$ and 2 ,
at $\lambda=4$ or 2 ,
then $x =1$ or $-1$
So maximum value of $x+\lambda=5$