Along with the other good advice you've already received...
Pay attention to what's going on when your child has a tantrum. Are you feeding the baby? Is it just before or after supper? Is it after school? Has she had a snack? Has she just returned from school? Keep a log of the tantrums for several days and look for a pattern.
Also, evaluate her eating/drinking habits. Does she drink sodas or sugary drinks? Does she eat what would be called a healthy diet (fruits, vegetables, protein) or does she eat processed snack foods or canned foods or fast foods primarily? Some kids react negatively to sugars, or processed artificial ingredients. And some can't handle caffeine or artificial sugars.
She may be experiencing hunger, if she doesn't eat much in school. Try a healthy snack after school that has some good fats and protein (a healthy, non-sugary yogurt, fruit, peanut butter on whole grain bread, cheese, but no candy or pre-packaged convenience stuff).
I do think an evaluation by your doctor could help. She may be low on a particular vitamin, or other simple things.
In the meantime, try using a physical sign with your hands (a "stop" signal like a crossing guard would use, or open your hand wide and then close it quickly) to give her a silent signal to calm down, and go to her room and do the screaming there. Don't react with your facial expression. Don't stop what you're doing. Don't let on that you're frustrated.
In a calm moment, try asking her if she knows when she's getting angry. How does she feel? Does she know she's about to explode in anger? Help her learn some relaxing breathing techniques, or help her realize she needs to leave the room.
Above all, don't react to her angry words. Tell her in a calm moment that your family doesn't say "shut up" or "I hate you" and that using those words will result in a loss of family time and mommy's attention. And of course, make sure that your language and that of your older children are calm, kind, and helpful. Talk to her about what you do when you're tired, or angry, or frustrated. You don't scream at your kids (hopefully), but you go take a relaxing shower or take a brief walk or count your blessings or think of something helpful, etc.