STEP-BY-STEP ANSWER:
Step 1: Prepare the bacterial smear on a microscope slide.
Step 2: Apply the primary stain (crystal violet) to color all bacteria.
Step 3: Add iodine, which forms a complex with the crystal violet, trapping it within the cell wall.
Step 4: Perform a decolorization step using alcohol or acetone; gram-positive bacteria retain the dye-complex due to thick peptidoglycan layers, while gram-negative bacteria lose the dye due to their thinner peptidoglycan layers.
Step 5: Counterstain with safranin, which stains the decolorized gram-negative bacteria pink.
Final Answer: Gram-positive bacteria appear purple and gram-negative bacteria appear pink under the microscope.