Through this video I document the solution for the challenge. The PUT method is used to take advantage of the API /api/BasketItems endpoint.
I make a HTTP request with the help of the cURL utility. The verb I use is PUT and as for the data I pass the following JSON.
'{"ProductId": 1, "BasketId": 1, "quantity": 1}'
In the above, the BasketID: 1 is not mine, but of a different client.