We　give　the　first　exact　and　stability　results　for　a　hypergraph　Turan　problem　with　infinitely　many　extremal　constructions　that　are　far　from　each　other　in　edit-distance.　This　includes　an　example　of　triple　systems　with　Turan　density　2/9,　thus　answering　some　questions　posed　by　the　third　and　fourth　authors　and　Reiher　about　the　feasible　region　of　hypergraphs.　Our　results　also　provide　extremal　constructions　whose　shadow　density　is　a　transcendental　number.　Our　novel　approach　is　to　construct　certain　multilinear　polynomials　that　attain　their　maximum　(in　the　standard　simplex)　on　a　line　segment　and　then　to　use　these　polynomials　to　define　an　operation　on　hypergraphs　that　gives　extremal　constructions.