The number of irrational roots of the equation $4x/(x^2+x+3)+5x/(x^2-5x+3)=-3/2$ is?

Let y = (x^2 + 3)/x, which transforms the equation into 4/(y + 1) + 5/(y - 5) = -3/2. Multiplying by 2(y + 1)(y - 5) and simplifying yields the quadratic equation 3y^2 - 2y - 35 = 0. Factoring this gives (3y + 7)(y - 5) = 0, so y = 5 or y = -7/3. For y = 5, the equation x^2 - 5x + 3 = 0 gives two real irrational roots. For y = -7/3, the equation 3x^2 + 7x + 3 = 0 has a positive discriminant of 13, giving another two real irrational roots, for a total of 4 irrational roots.

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