Given,
Mean of observation is 15,
So 20∑xi=15
⇒∑xi=15×20=300⋯(i)
Also variance is given as 9
So, 20∑xi2−(20∑xi)2=9⋯(ii)
⇒20∑xi2−(15)2=9⋯(ii)
⇒∑xi2=234×20=4680
Given mean of (x1+α)2,(x2+α)2,…,(x20+α)2 is 178
So, 20∑(xi+α)2=178⇒∑(xi+α)2=3560
⇒∑xi2+2α∑xi+∑α2=3560
⇒4680+600α+20α2=3560
⇒α2+30α+56=0
⇒(α+28)(α+2)=0
α=−2,−28
Square of maximum value of α is 4