We consider the set of functions given by

We consider the set of functions given by 

$\int a (x) = (x-a)^2 + (4a^2-12a)$

 the equation with real parameter a.

a) Determine all those values for a for which the function $\int a (x)$ has exactly two different zeros.

b) Among all the values for a for which the function $\int a (x)$ has exactly two different zeros, determine those for which the distance between these zeros is maximum.