You're more or less right on a priori -- though it's important that it doesn't mean, for Kant, that it can't be known from experience. That is, you can learn an a priori expression from experience, but it has to be the sort where you don't need experience to know it.
Synthetic propositions are those that add truth value beyond that given in the terms used in the proposition. So a statement like "That cat is red" is synthetic, since the term 'cat' doesn't mean the term 'red'. Kant claims (disagreeing with Hume) that there are synthetic a priori prepositions; an example of this is "2+2=4", since '2+2' doesn't mean 4.
You're confusing 'synthetic' with 'a posteriori'. For Hume they are identical, but for Kant, they are.
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"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."
"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."
-- Friedrich Nietzsche
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