Time/Space Requirements of Verifying or Falsifying a First-Order Statement
The theory of real closed fields (RCF) is complete for the first order theory of real numbers in the language you described. Therefore it is equivalent to checking if RCF proves the formula.

By Tarski's quantifier elimination for RCF this can be computed. Tarski's algorithms complexity is non-elementary and there is a doubly exponential lowerbound for the problem.

A more efficient algorithm than Tarski's algorithm is given by Saugata Basu "An Improved Algorithm for Quantifier Elimination Over Real Closed Fields", FOCS 1997 (which seems to be almost optimal, see Theorem 1) (a draft of the paper is available here). 

Also check Saugata Basu, "New Results on Quantifier Elimination Over Real Closed Fields and Applications to Constraint Databases", Journal of the ACM, 1999.