Partial fraction decomposition means that you need to consider the final result and try to write it as a sum or difference of initial irreducible fractions.

Considering the fraction `(2x+1)/((x+1)(x+2)),` you need to write the irreducible fractions such that:

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

Notice that you do not know the numerators, hence, you need to find them such that:

`2x + 1 = A(x+2) + B(x+1)` (the fractions from the right are brought to a common denominator)

`2x + 1 = Ax + 2A + Bx + B`

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

Equating the coefficients of like powers yields:

`A+B = 2 =gt B=2-A`

`2A+B = 1 =gt 2A + 2 - A = 1 =gt A = -1`

B = 3

**Hence, decomposing the fraction yields `(2x+1)/((x+1)(x+2)) = -1/(x+1) + 3/(x+2).` **