Hydrazine burns in oxygen. How many bonds are broken and how many bonds are formed in a reaction?

The correct answer depends upon the products and you didn't specify those. The following is one set of products.

H2N-NH2 + O2 ==> HOH + HOH + N2
You count the bonds broken (H-N, H-N, H-N, H-N) and bonds formed (H-O, H-O, H-O, H-O). Check that.