Concept:
Congenital hypothyroidism (cretinism) is one of the most common treatable and preventable causes of neurocognitive impairment and intellectual disability in children.
The classic constellation of clinical findings in an untreated infant includes gross motor and global developmental delay, coarse facial features, macroglossia (large protruding tongue), large persistent umbilical hernia, hypotonia, open posterior fontanelle, prolonged neonatal jaundice, and hoarse cry.
Explanation:
• Thyroid hormones ($T_3$ and $T_4$) are critically required for early neurogenesis, myelination, dendritic branching, and skeletal maturation during the first few months of postnatal life.
• Untreated thyroid deficiency leads to accumulation of glycosaminoglycans within interstitial spaces, causing typical coarse facial features, myxedematous skin, and a large protruding tongue.
• Abdominal muscle hypotonia combined with delayed closure of the umbilical ring directly leads to a prominent umbilical hernia.
• The definitive, quickest, and most essential initial diagnostic test is the measurement of serum Thyroid Stimulating Hormone (TSH) and Free Thyroxine (FT4).
• In primary congenital hypothyroidism, laboratory evaluation characteristically shows significantly elevated serum TSH with low or undetectable Free T4 levels.
• Prompt confirmation and immediate initiation of oral Levothyroxine therapy (10--15 $\mu\text{g/kg/day}$) are crucial to prevent irreversible neurodevelopmental sequelae.
Final Answer:
The first and most critical investigation in an infant presenting with developmental delay, coarse facies, macroglossia, and umbilical hernia is serum TSH and Free T4.