You will need: a sharp knife, a steady hand, and patience.
First carve 5 equally sized 1/4 spheres out of carrot.
Next, following the curve of one of the notches you have just cut out, cut a slit just beneath the edge of the cut. Then, following the same line, cut another slit a centimeter down from the first slit connecting the two so it will slid out like a puzzle piece.
Cut the end of the "petal" at an angle to resemble a real flower.
Next cut a triangular piece out from the next spherical notch and then cut the two slits out like the last step. Continue all the way around until you reach your starting point.
When you reach your starting point you will then angle your cuts beneath the cuts you have already made, making sure not to disturb the previous cuts(adjust all cuts for the size of the carrot, and how big you want your flower to be).
When approaching the top of your carrot move inwards and make much smaller cuts but using the same process.
Like petals of a flower come from the inside-out we are coming from the outside-in to make the carrot feel real to the onlooker.