Partial fractions:
Partial fractions is a technique used to simplify complex rational expressions by breaking them down into simpler fractions. It involves decomposing the rational expression into a sum of simpler fractions, each with a denominator that is a factor of the original denominator.

Steps to resolve the rational expression:
To resolve the rational expression 4x³-7x²+5x-4 / (x²-1)(x²-3x-2) into partial fractions, follow these steps:

Step 1: Factorize the denominator:
First, factorize the denominator (x²-1)(x²-3x-2) to make it easier to work with.

The factorization of (x²-1) can be done as (x+1)(x-1), and the factorization of (x²-3x-2) as (x-2)(x-1).

So, the denominator (x²-1)(x²-3x-2) can be written as (x+1)(x-1)(x-2)(x-1).

Step 2: Decompose the rational expression:
The rational expression can be written as a sum of simpler fractions with the factors of the denominator as their respective denominators, and unknown constants as their numerators.

4x³-7x²+5x-4 / (x+1)(x-1)(x-2)(x-1) can be decomposed as A/(x+1) + B/(x-1) + C/(x-2) + D/(x-1).

Here, A, B, C, and D are unknown constants that need to be determined.

Step 3: Find the unknown constants:
To find the values of the unknown constants A, B, C, and D, we need to find the common denominator of the fractions on the right-hand side.

The common denominator can be found by multiplying all the denominators together, which gives us (x+1)(x-1)(x-2)(x-1).

Multiplying the denominators with the respective numerators, we get:
4x³-7x²+5x-4 = A(x-1)(x-2)(x-1) + B(x+1)(x-2)(x-1) + C(x+1)(x-1)(x-1) + D(x+1)(x-1)(x-2).

Step 4: Solve for the unknown constants:
To solve for the unknown constants A, B, C, and D, we substitute values for x that make some of the terms zero. This allows us to solve for the remaining unknown constants.

Substituting x = 1, the terms with (x-1) in the denominators become zero. This leaves us with:
4(1)³-7(1)²+5(1)-4 = A(1-1)(1-2)(1-1) + B(1+1)(1-2)(1-1) + C(1+1)(1-1)(1-1) + D(1+1)(1-1)(1-2).

Simplifying the equation, we get:
-6 = -3A.

From this equation,

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