OK,

(1) The product of the ages = 36

(2) The sum of their ages is the number of cars in my street.

(3) And to determine their ages

Given the fact that I know the # cars in my street and I still say the first half of sentence 3, implicates that from the product and sum alone one cannot determine the ages.

1 1 36 sum 38

1 2 18 sum 21

1 3 12 sum 16

1 4 9 sum 14

1 6 6 sum 13

2 2 9 sum 13

2 3 6 sum 11

3 3 4 sum 10

given that sum and product is not enough to determine leaves only 2 options.

1 6 6 sum 13

2 2 9 sum 13

and the youngest was recently ill, so ....