(2x^2-x+4)/ x^3+4x

Factor the denominator and then the fraction may be written in the form:

(2x^{2} – x + 4)/(x^{3} + 4x) = (2x^{2} – x + 4)/ [x(x^{2} + 4)] = A/x + (Bx + C)/(x^{2} + 4)

Multiply both sides by x^{3} + 4x. Then for all x we have 2x^{2} – x + 4 = A(x^{2} + 4) + (Bx + C)x .

Substitute x = 0 into the previous identity. Then 4 = 4A and A = 1.

Write the coefficients at x and at x^{2}: –1 = C, 2 = A + B, and B = 2 – A = 1.

Answer: (2x^{2} – x + 4)/(x^{3} + 4x) =1/x + (x – 1)/(x^{2} + 4).

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