Pilgrim’s Pride, which operates a plant in Gainesville, launched a takeover bid for Hillshire Brands on Tuesday that would expand its business beyond fresh and frozen chicken into a variety of packaged meats, including Ball Park hot dogs and Jimmy Dean sausages. The $5.58 billion offer could derail Hillshire’s plans to buy Pinnacle Foods, which makes Birds Eye frozen vegetables, Duncan Hines cake mixes and Hungry Man frozen dinners. Hillshire had announced the $4.23 billion deal earlier this month as a way to expand into different parts of the supermarket.