What is the partial fraction decomposition of (5x+1)/(x^2+x-2) ?

First thing, we'll write the denominator as a product of
linear factors. For this reason, we'll determine it's
roots.

We notice that the sum of roots is -1 and the
product is -2, then the roots are x1 = 1 and x2 =- 2.

We
can re-write the denominator as:

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

Since the factors from denominator are of the
form (x - a), then we'll write the partial fractions
as:

(5x + 1)/(x-1)(x+2) = A/(x-1) +
B/(x+2)

(5x + 1) = A(x+2) +
B(x-1)

We'll remove the
brackets:

5x + 1 = Ax + 2A + Bx -
B

5x + 1 = x(A+B) + 2A -
B

Comparing, we'll get the
system:

A + B = 5

2A - B =
1

Adding the equations above, we'll
get:

A+B+2A-B = 5+1

3A =
6

A = 2

2 + B = 5 => B
= 3

The complete partial fraction
decomposition is (5x + 1)/(x-1)(x+2) = 2/(x-1) +
3/(x+2)