Of course science and religion can mix and they should!
Let me quote Abdulbaha, son of the founder of the Bahai religion, a growing religious and social movement with more than six million followers:
If religious beliefs and opinions are found contrary to the standards of science, they are mere superstitions and imaginations; for the antithesis of knowledge is ignorance, and the child of ignorance is superstition. Unquestionably there must be agreement between true religion and science. If a question be found contrary to reason, faith and belief in it are impossible, and there is no outcome but wavering and vacillation.
Quite a strong statement for being from a major religious leader a hundred years ago. Also:
This gift [intelligence and reasoning] giveth man the power to discern the truth in all things, leadeth him to that which is right, and helpeth him to discover the secrets of creation
Religion and science are the two wings upon which man's intelligence can soar into the heights, with which the human soul can progress. It is not possible to fly with one wing alone! Should a man try to fly with the wing of religion alone he would quickly fall into the quagmire of superstition, whilst on the other hand, with the wing of science alone he would also make no progress, but fall into the despairing slough of materialism.
The only reason that science and religion doesn't seem to mix is that too many religious leaders stick to their dogmas and traditions even in face of human and scientific progress. Religions role in this world is to develop and foster spirituality, morality and selflessness so we can create a fair and just society and it can only do so if it keeps evolving and improving with new knowledge and understandings. Christianity developed and changed a lot in the first few hundred years after Jesus with doctrines and writings being added and removed at a high pace. Why are so many churches of today so hellbent on sticking exactly to the way things earlier were? It's simply not healthy.
Ps. I'm not officially a Bahai, but I consider myself a "friend of the faith".