This is a beginners question. I and reading the book "Introduction to Computer Theory" by Daniel Cohen. But I end up with confusion regarding simplification of regular expressions and finite automata. I want to create an FA for the regular expression
$\qquad \displaystyle (a+b)^* (ab+ba)^+a^+\;.$
My first question is that how we can simplify this expression? Can we we write the middle part as $(ab+ba)(ab+ba)^*$? will this simplify the expression?
My second question is whether the automaton given below is equivalent to this regular expression? If not, what is the mistake?
This is not a homework but i want to learn this basic example. And please bear me as a beginner.