Clearly $4350$ is divisible by $5$, since the last two digits are $50$. It is also divisible by $3$, since the sum of the digits is divisible by $3$. Dividing by $150$ gives a quotient of $29$, which is prime, so the prime factorization is $4350 = 2 \times 3 \times 5^2 \times 29$, and the largest prime factor is $29$.