Use Descartes' Rule of Signs to find the number of possible positive real roots and the number of possible negative real roots for the function f(x) = x4 + 2x3 - 3x2 - 8x - 4.

For positive roots, read the signs of the coefficients of f(x): +, +, −, −, −. There is 1 sign change, so there is 1 possible positive real root. For negative roots, substitute −x to get f(−x) = x⁴ − 2x³ − 3x² + 8x − 4, whose signs are +, −, −, +, −. There are 3 sign changes, so the number of negative real roots can be 3 or 1. This matches choice A: positive 1; negative 3 or 1.

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