5x+1/x^2(x^2+4) decompose into partial fractions

Answer 1

(5x+1) / x^2(x^2+4) = A/x + B/x^2 + (Cx+D)/(x^2+4
5x+1 = Ax(x^2+4) + B(x^2+4) + (Cx+D)x^2
that gives you
Ax^3 + 4Ax + Bx^2 + 4B + Cx^3 + Dx^2 = 5x+1
(A+C)x^3 + (B+D)x^2 + 4Ax + 4B = 5x+1
4A = 5
4B = 1
A+C=0
B+D=0
solve those and you wind up with
(5x+1) / x^2(x^2+4) = (5/4)/x + (1/4)/x^2 + ((-5x-1)/4)/(x^2+4)
or
5/(4x) + 1/(4x^2) - (5x+1)/(4(x^2+4))

Answer 2

3 and 8

Answer 3

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