Can you use both an U.S. and Mexican passport during while traveling?

[Sputnik]

I hold both an American passport and a Mexican passport (dual citizenship). I hear you should enter Mexico with your Mexican passport and when returning to the US, use the American passport. Also, I'd like to go to Cuba on my Mexican passport (form Mexico) and re-enter the US, as an American..Does anyone here know how this works? Mexico does allow dual citizenship in cases of naturalization abroad (The US, however, doesn't)...

Merci..