Let the common factor be x−a for both f(x) and g(x).
Now x−a is a factor
So, f(a)=a2+15a+m=0 a2+15a+m=0 a2=−15a−m....(i)
Also, g(a)=a2+16a+n=0 a2+16a+n=0 a2=−16a−n...(ii)
From (i) and (ii), −15a−m=−16a−n ⇒m−n=a ⇒(m−n)2=a2 ⇒(m−n)2=−15a−m or (m−n)2=−16a−n ⇒(m−n)2=−15(m−n)−m or (m−n)2=−16(m−n)−n ⇒(m−n)2=−15m+15n−m or (m−n)2=−16m+16n−n ⇒(m−n)2=−16m+15n or (m−n)2=−16m+15n