Let the common factor be x−a for both f(x) and g(x)
Now, x−a is a factor
So, f(a)=a2+21a+m=0 a2+21a+m=0 a2=−21a−m...(i)
Also, g(a)=a2+22a+n=0 a2+22a+n=0 a2=−22a−n...(ii)
From (i) and (ii), −21a−m=−22a−n ⇒m−n=a⇒(m−n)2=a2 ⇒(m−n)2=−21a−m or (m−n)2=−22a−n ⇒(m−n)2=−21(m−n)−m or (m−n)2=−22(m−n)−n ⇒(m−n)2=−21m+21n−m or (m−n)2=−22m+22n−n ⇒(m−n)2=−22m+21n or (m−n)2=−22m+21n