Separate the fraction 1/(x^2+x) into partial ratios?

First, we'll factorize by x the
denominator:

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

We notice that the denominator of the right side
ratio is the least common denominator of 2 irreducible
ratios.

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

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

We'll multiply the ratio A/x by (x+1) and we'll
multiply the ratio B/(x+1) by x.

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

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

1 = A(x+1) +
Bx

We'll remove the
brackets:

1 = Ax + A +
Bx

We'll factorize by x to the right
side:

1 = x(A+B) + A

If the
expressions from both sides are equivalent, the correspondent coefficients are
equal.

A+B = 0

A =
1

1 + B = 0

B =
-1

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

1/x(x+1) = 1/x -
1/(x+1)