Clearly for two quadratic equations x^2+m*x+n=o and x^2+m'*x+n'=o to have identical couples of roots m must be equal to m' and n=n'.
In our case A=B.
So x1+x2=A and x1*x2=A
The above set has only one solution in integers: A=B=4: x1=x2=2.
'Prove that there are no others."-
AFTER FINDING ALL SOLUTIONS this is an unnecessary/redundant demand and should be avoided in the future.
n.b. please see my next comment
Edited on October 18, 2016, 7:33 pm