Question

write the partial fraction decomposition step by step

((5x-1)/(x^2+4)) - ((4)/(x^2+4)^2)

Answer 1

Correction:
((5x-1)/(x^2+4)) - ((4/(x^2+4)^2))

Answer 2

I'm so sorry. That was the answer to the problem that I'm trying to solve.

The actual problem is
(5x^3-x^2+20t-8)/((x^2+4)^2)
I would like to know how to solve it step by step

Answer 3

let (5x^3 - x^2 + 20x - 8)/(x^2+4)^2
= (Ax + B)/(x^2 + 4) + C/(x^2+4)^2
or
= ( (Ax+b)(x^2+4) + C )/(x^2=4)^2
= ( Ax^3 + 4Ax + Bx^2 + 4B+C)/(x^2+4)^2

matching up coefficients of the original numerator ...
A = 5
B = -1
4A = 20 ---> confirms A=5
4B+C = -8
-4+C = -8
C = -4

so the de-composition is
(5x - 1)/(x^2+4) - 4/(x^2+4)^2

Answer 4

In general, it would be

(Ax+B)/(x^2+4) + (Cx+D)/(x^2+4)^2

In this case, D just happens to be zero.