How can I write an Egos Logos and Pathos essay on the acquisition of Texas and Florida?

I think you mean "ethos," logos and pathos. Ethos is an ethical appeal. Logos is a logical appeal and pathos is an emotional appeal. So you would need to write an essay which appeals to the audience's sense of ethics (right and wrong), their sense of logic (reasoning) and emotion (make them feel sadness or hope).