How to decompose in partial fractions 1/(2x^2+4x)?

First, we'll factorize by 2x the
denominator:

1/(2x^2+4x) =
1/2x(x+2)

We'll suppose that the fraction 1/2x(x+2) is the
result of addition or subtraction of 2 irreducible partial
fractions:

1/2x(x+2) = A/2x + B/(x+2)
(1)

We'll multiply the ratio A/2x by (x+2) and we'll
multiply the ratio B/(x+2) by 2x.

1/2x(x+2)= [A(x+2) +
2Bx]/2x(x+2)

Since the denominators of both sides are
matching, we'll write the numerators, only.

1 = A(x+2) +
2Bx

We'll remove the
brackets:

1 = Ax + 2A +
2Bx

We'll factorize by x to the right
side:

1 = x(A+2B) +
2A

Comparing, we'll get:

A+2B
= 0

2A = 1 => A =
1/2

1/2 + 2B = 0

B =
-1/4

We'll substitute A and B into the expression
(1):

The result of decomposition into partial
fractions is: 1/2x(x+2) = 1/4x - 1/4(x+2)