Find the domain of f of x equals negative square root of the quantity two x plus four, plus 3?

You always find the domain of these by setting what's under the radical sign equal to 0 and solving for x.  In this case, 2x+4=0.  2x= -4 and x = -2.  If you graph it, you will see that it is true.  The negative sign out front means that the graph is "upside down" from its positive parent graph, and the "+3" means that it is moved up from the origin, but the domain is found by setting the radicand equal to 0 and solving for x.  Domain: [-2, infinity)