We can use the quadratic formula to find the roots (zeroes), using just the expression under the square root sign. For rational zeroes we need a perfect square. B2-4AC must be positive for real zeroes and it must evaluate to a perfect square, where A=2, B=a+1, C=-(3a2-4a+b).
B2-4AC=a2+2a+1+8(3a2-4a+b)=
25a2-30a+8b+1.
This has to be a perfect square which is of the form (5a-d)2=25a2-10ad+d2, where d has to be determined.
So, 10ad must be equal to 30a, making d=3. d2=9 so 8b+1=9, b=1. So 5a-3 is the square root which is rational when a is any rational number and b=1.