Awoman gives birth to quadruplets or four infants. two of her children are identical twins. the other two are fraternal twins. how could this have happened

To explain this, let's go back to the fertilization:
The identical twins are formed by an association of one spermatozoid and one ovule to form an egg. this egg will start growing and suddenly divide into 2 eggs, to give afterward 2 individual with the same genes.
The fraternal twin comes from two different spermatozoids which will fertilize one ovule each, and give 2 different individuals sharing half of their genes.

In this case, both phenomena happened. It means there were three spermatozoids fertilizing one ovule each. And one of the eggs divided into 2 eggs. it ended by 2 eggs which are different and 2 eggs which came from the same one.