First we'll find the number of teams of 4 from the 36 children without
restrictions of gender.  Then we'll subtract the number with no boys
and the number with just 1 boy.

1.  The number of teams with no restriction of gender.

That's 36C4 = 58905 teams with no restrictions of gender.

2.  The number of teams with no boys, which means 4 girls:

That's 19C4 = 3876 teams of all girls, no boys, which we must subtract.

3.  The number of teams with exactly 1 boy and 3 girls:

We can choose the 1 boy 17C1 = 17 ways.
Then we can choose the 3 girls 19C3 = 969 ways.
Thats 17(969) = 16473 ways, which we also must subtract.

Final answer = 58905-3876-16473 = 38556 teams with at least 2 boys.

Edwin