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The real expressions that are entered will have numbers where the discussion so far has talked about a,b,c.
Will the next character after the optional leading number be x?

All of that should now...

What characters make up an <expr>? Is there an implied * between a and x in ax?
What characters make a <sign>
What separates an <expr> from a <sign>

The choices are one of these three?
<expr>
<expr> <sign> <expr>
<expr> <sign> <expr> <sign> <expr>

Then the code needs to be able to parse out <expr> and <sign> from the String that was entered.

What is the syntax of the full expression?
Will it always have three expressions separated by sign operators? Could some of the parts be omitted? say two expressions and one sign operator?
<exp>...