My impression is that when religious people say they can't believe in science, what they generally mean is that they are uncomfortable with evolution, the earth being 4.6 billion years old, and the creation. I think most of us were trained up in Sunday School to believe that God created the earth in six literal days, that it all happened "magically" (not quite the right term, but hopefully you get my meaning) just because God said so, and that Adam and Eve were made from scratch by God. So when the idea of an old earth and evolution are brought up it seems to conflict with our religion and we feel like we have to reject it. That doesn't mean we object to the science that has been used to develop computers, cell phones, advanced medical treatments, faster and more efficient cars, or flushing toilets.
I admit that there seem to be conflicts there. Over time I've gotten to the point where I don't have a problem with the earth being old or with it taking more than six literal days for it to be created. Considering the history behind where we got the Bible, I don't think we necessarily have to take the word "day" literally as 24 hours. When I consider the creation of the earth as told by geology, I think it's pretty amazing that God and Jesus Christ knew all that and much more and that they planned everything so that a planet would be created where humans could live.
However, I do still have some problems with the theory of evolution. I do think there is truth to it. I think there is plenty of evidence to support it. But it does rub wrong with my conception of the creation of Adam and Eve. Evidence from the geologic record seems to indicate that there was a progression from ape-type creatures to humanoids to humans, and there are things about that idea that I don't really like. I believe that there is something special about the human species, namely, that we are God's literal spirit children. I'm still stuck on the ideas I picked up in church to some extent, and that makes me inclined to think that God created Adam and Eve from scratch. With modern science it seems conceivable that someone could create life from scratch (in a test tube, if you will). But if He did create them through the process of evolution, that would seem to mean that Adam and Eve had parents who didn't quite make the cut. Does that mean their spirits weren't quite good enough to belong to human bodies and be God's children? That seems harsh to me and I don't really like it.
At this point my conclusion is that we don't know enough. Our understanding of spirits is pretty minimal, and I think that's one reason why it's hard to fit the things we know about spirits with the theory of evolution. I'm okay with waiting until we have more understanding. Scientific evidence suggests that there is truth to evolution. I think we should continue to study and teach it because if we understand evolution we might be able to use that understanding to bless our lives. But I also know that my faith is true and real. My personal experiences have proven that to me.
I think the best way to receive answers is to continue to obey God's commandments. Many answers to our gospel questions don't come immediately. They come when we continue to be faithful and obedient. Gradually (through study and experience) we learn and understand concepts until eventually we are prepared to understand the question we started with long before. If we continue to learn and grow in the gospel, we will gain further light and knowledge concerning spirits, the creation, the nature of God, and other gospel principles that are important in understanding why we are here and how we got here. If we reject God and refuse to accept the scriptures, we will never experience that growth. So, interestingly, it seems to me that in order to fully understand science we have to also accept religion, because at some point the two will meet in the middle and the issues will be resolved. If we reject religion, the two will never meet and we will miss out on important parts of the whole picture.