Final answer:
To decompose the given expression into partial fractions, factor the denominator completely and write each factor as a distinct fraction.
Step-by-step explanation:
To decompose the given expression into partial fractions, we need to factor the denominator completely. The denominator, (3x+2)(x+1)³, is already fully factored.
Let's find the partial fraction decomposition for each factor of the denominator:
The first factor, 3x+2, gives us A/(3x+2) where A is a constant.
The second factor, x+1, gives us B/(x+1) where B is a constant.
The third factor, x+1, gives us C/(x+1) where C is a constant.
Putting it all together, the decomposed expression is: A/(3x+2) + B/(x+1) + C/(x+1)³.