Examining the degrees of polynomials in ∃x ∈ ]a, b[, P = 0 and Q₁ > 0 and ... Q_k > 0

• The nice case is when degrees in the variable x of the polynomial Q_i are strictly smaller that than degree in the variable x of the polynomial P, that is, when degree_x(Q₁) < degree_x(P)... and degree_x(Q_k)< degree_x(P).
  Those formulas are ok:
  
  ∃x ∈ ]a, b[, (x² - x = 0) and (x + 2 > 0)
  
  ∃x ∈ ]a, b[, (x⁵= 0) and (x + 1 > 0) and (x -5 > 0)
  click here.
• Otherwise, we need to simplify the formula in order to have degree_x(Q₁) < degree_x(P)... and degree_x(Q_k)< degree_x(P).
  Those formulas need to be simplified:
  ∃x ∈ ]a, b[, (x + 6 = 0) and (x² + 9x - 5 > 0)
  
  ∃x ∈ ]a, b[, (x³ = 0) and (x³ > 0) and (x > 0)
  Let us simplify the formula!.