Final answer:
To solve the given rational equation with the specified conditions, one must find a common denominator, combine terms, and then potentially use the quadratic formula or other methods if the resulting polynomial is higher than degree two.
Step-by-step explanation:
To solve the equation 2x/(x-3) + 1/(2x+3) + (3x+9)/((x-3)(2x+3)) = 0 with the condition that x ≠ 3 and x ≠ -3/2, we need to find a common denominator and simplify the expression. The common denominator in this case is (x-3)(2x+3).
Once we have the common denominator, we can combine the terms and form a single rational expression equal to zero. Next, we'll multiply through by the common denominator to get rid of the fractions and obtain a polynomial equation. This polynomial could be quadratic or have a higher degree, depending on the terms involved. If it's a quadratic equation, we can solve for x using the quadratic formula, which is x=(-b±√(b²-4ac))/(2a) for an equation of the form ax²+bx+c=0.
If the polynomial has a degree higher than two, we cannot use the quadratic formula directly. Instead, we might need to use polynomial long division, synthetic division, factoring, or numerical methods such as using a graphing calculator to find the zeros.