The question here is asking us to rewrite the complicated fraction (2x^3 - 10x + 4) / ((x + 1) * (x - 3)^2) as a sum of simpler fractions, which is a process called partial fraction decomposition.
The procedure of partial fraction decomposition involves expressing the given fraction as the sum of simpler fractions. The denominator of the given fraction is ((x + 1) * (x - 3)^2). From this, we can see that the factors are (x + 1) and (x - 3). Furthermore, (x - 3) is squared, indicating that it will be in two of the fractions in the decomposition.
Since the denominator of the original, more complicated, fraction is the product of two factors, (x + 1) and (x - 3)^2, our objective is to write it as the sum of fractions, each with only one of these factors in the denominator. This helps in integrating or further simplification, if needed.
Therefore, we will express the original fraction in the following general form:
A / (x + 1) + B / (x - 3) + C / (x - 3)^2
Here, A, B, and C are constants that we would find if we were to solve this problem completely. Therefore, the partial fraction decomposition of (2x^3 - 10x + 4) / ((x + 1) * (x - 3)^2) is
A / (x + 1) + B / (x - 3) + C / (x - 3)^2
We can't go further into this without the actual values. The values of these constants, A, B, and C, can be calculated by forming sets of linear equations and solving for them, if the problem asked for a more specific result.