Step 1: Identify the clinical picture.
A child with fever, swollen neck glands, and a gray membrane on the tonsil that spreads onto the anterior pillar is a classic picture of diphtheria, caused by Corynebacterium diphtheriae. The membrane tends to bleed if scraped, which helps tell it apart from ordinary tonsillitis.
Step 2: Recall how this organism is cultured.
Corynebacterium diphtheriae grows on a few special media. Loeffler's serum slope is an enriched, non-selective medium made of coagulated serum, and it lets the bacteria grow fast and show their shape clearly.
Step 3: Explain why Loeffler's slope gives a rapid answer.
After only 6 to 8 hours, a smear from a Loeffler's slope culture, stained with Albert's or Neisser's stain, shows the club-shaped bacilli with metachromatic (volutin) granules at their ends. This lets the lab give a quick presumptive report while the definitive culture on tellurite medium, which takes 24 to 48 hours and shows black or grayish colonies, is still growing.
Step 4: Rule out the other media.
(1) Nutrient agar: Too plain, it does not favor this organism and gives no early clue to its identity.
(2) Blood agar: Useful for many throat pathogens like Streptococcus pyogenes, but it does not speed up identification of C. diphtheriae.
(4) Lowenstein-Jensen medium: This is used to grow Mycobacterium tuberculosis, not C. diphtheriae.
Final Answer:
Loeffler's serum slope is the medium of choice for a rapid, early look at the diphtheria bacillus.
\[ \boxed{\text{Loeffler's serum slope}} \]